✽A quick TLDR to help in understanding Servants. Any information here is optional to read, I just want to provide it because it can be hard to discern Fate characters if you are not in the fandom.
――― What is a Holy Grail? Accounts of the Holy Grail differ in the entirety of Nasu works, but for the most part. To understand it I will simply explain the original use for it: The Grail is a relic or a magical item that was originally used to summon Grand Level Servants. To defeat Beast level calamities back in the day when the world had more potent magic in it.
It was not meant to be utilized ever again, however, magus families later on. In the future, sometime around 1500s or w.e, decided that in order to reach the root. They could do so, by more or less copying this system of ritualistic summoning, to sacrifice souls and more or less reach the root this way.
From there - a bunch of spin-offs happen. Blablablablalblab shit happens a lot.
The Grail does not inherently grant wishes, it more or less establishes a connection to the 'Root,' which, then if you are smart enough... You can do something with it, and THUS, create the wish that you want out of it.
The reason servants were needed, is because humans cannot touch the grail or they will die.
A Grail ultimately is a very ridiculous powerful mana source.
――― What is a Servant? An over glorified Familiar, more or less. I don't recall and I dont really want to dive looking into it too much, but basically, the Grail establishes a connection between the Material plane and wherever the fuck Nasu decided the Throne of Heroes exists within. The Throne of Heroes is merely a hub, basically, where notable beings are recorded and kept within - kind of like Valhalla in Norse Mythology, but isn't actually Valhalla. So do not confuse the two.
Servants are predisposed when summoned by the grail, in being altered by the perception of how Humanity perceives them. This is because they technically also hail from the collective unconscious, and thus are again, predisposed to humanities contingencies/understanding.
――― Why are Servants so Powerful? Because they were literally originally used to fight manifestations of Human Calamities. That was their original purpose. Additionally, they are connected to the root/are potent figures made of magic.
――― What are Command Seals? Tldr, the thing that pins and binds a servant into reality. With this, they are commanded to obey their master or otherwise, be subjected to possibly their Master using these seals against them. This is a contract more or less, between Servant and Master. It is usually in 3 seals, that are tattooed on the master's hand.
――― What is the Master & Servant Relationship Like? Almost parasitic, the servant uses their Master's mana output to stabilize their existence in the material realm. They can communicate by mind, and memory. While sharing past moments with their Master as they sleep. This is not controlled by the servant or master. Additionally, the Command seals given to the Master can be used to strengthen or Weaken their servant.
――― What constitutes a familiar or Servant in Fate? Actual information about Familiar is here. I will copy paste the more important part: Ghost Liner: Boundary Record Belt (境界記録帯ゴーストライナー, Kyōkai Kiroku TaiGōsuto Rainā?) is the greatest type of Familiar, that is, Servants Strictly speaking, they are not a Familiar and rather fall more on the side of a summoned being. Servants are called Familiars only due to some similarities (spiritual connection, dependence on Magical Energy) and for the sake of formalities. They need to be put in a class of their own however, as it would be impossible to summon Heroic Spirits without the help of the Holy Grail, and there is no need to use something as powerful as Command Spells to control regular Familiars.
The Important note here, is that Servants Require Magical Energy to exist as well as basically a pin to bind them onto the world(Which is a master). Otherwise, they risk more or less dispersing into the atmosphere and ceasing to exist. When they pin themself to a master, they consume their mana. Which is in a lot of ways their life force as well.
However, Servants canonically can exist and manifest, as long as they continuously consume mana in some other form. Which hilariously can be sexual or nonsexual - like just holding hands, casually making out, just simply eating food, eating.... Weird things from each other... Killing humans and eating their life force, and etc.
Additionally: Although Servants are beings of Ether that do not get energy from oxygen, they are given forms similar to air-breathing lifeforms, allowing them to "carry out pulmonary respiration as a land-type endoskeleton life form." Though they do not require oxygen to function, the lack of air from suffocation can halt their Magical Energy cycle. Though they can likely endure more than a human, they will eventually reach their limit if enough time passes. Differing from the normal blood of humans, the blood of Servants is the Magical Energy that drives their Saint Graph. You can read more here.
――― What Is Chaldea? Really quickly a TLDR - a place that intended to solve time shenanigan problems and keep shit intact or whatever. The system they use to summon spirits is very different from the Holy Grail, and does not whatsoever follow the same rules that the actual Holy Grail Follows. That is the only important thing you need to know.
――― What is the Moon Cell? Basically, a very over-glorified super computer. That lives in the moon, it was created a long time ago. Before humanity and life existed on Earth. It has been watching over and recording everything on earth since the beginning of time. 4,000 years ago, its data was destroyed because of the battle with Sefar. So the data currently only has about 4,000 years worth.
――― What are Class Skills? Skills that typically every class has, with some differences in levels. Or additional modifiers depending on the character it is. These are almost always Passive Skills.
――― What is a Reality Marble? TLDR, it is different per person but for the most part, it literally can be summoned up as a personal Pocket Dimension lmao.
――― What is a Spirit Origin? The body or main essence of a Servant. If you break the Spirit Origin, you destroy the Servant themself. “The Spiritual Core is described as "their center, their heart, their CPU" that dictates their every function.” - Here
――― What is a Noble Phantasm? An ultimate form of attack, requiring a disgusting amount of magical energy to expend. It is named and should always more or less be in essence, the character's entire story/meaning/conception in some way. Revealing ones noble phantasm, often is revealing ones true name.
――― Far Side of the moon? In the moon cell there are two sides, the far side is basically where nothingness is. Junk. etc. Void space.
――― Near Side of the moon Basically… The Material plane? Time flows naturally here. This is the place where you want to be.
――― Imaginary Number Space (void space) A layer or texture of the World (specific to world), think similarly to how DND uses its Plane system. The Moon has its own, called the Far side of the moon. This is the one primarily BB has control over. This is a space of area that is outside of all realities, almost as if it is the opposite of reality itself. Going into this is separating yourself from the reality you left from. When you come back out of it, you come back at an undetermined amount of time that can pass in the PRESENT reality. If someone other than BB were to go in here, without a (compass or Zero Sail) you likely would get lost. Grow Mad. Die horribly. This is the place where the Moon Cell dumps all Malignant Data.
――― What is a Singularity? Tldr; A time disparity that if not solved and closed, can ruin the timeline currently established. That is the short end of it, if a Singularity is solved - fixed. Usually when the Holy Grail is obtained (the sort of the singularity creation) then the singularity will simply just disappear. When it is closed, usually time anomalies happen because of issues created by the singularity.
――― What are Gods Authority? Tldr; In essence, these are authorities given only to God characters or characters associated with Gods. It basically means, what they speak is what will happen. "What I say, becomes so." Usually, the authority only works in and with what they are associated with as a God, for instance in BB's case. As she has Pele's God Core, she wanted the Hawaii islands to be merged together - so they were.

✽ God Authorities
List of Gods Consumed: Pele, Nyarlathotep, Goddess of Çatalhöyük, Tiamat, Cybele, Ishtar, Inanna, Anat, Ashtart, Gaia, Hera, Artemis, Aphrodite, Demeter, & Athena.
――― What is an Authority? Again: "What I say, becomes so."
――― Pele Creation of Luluhawa - Control the lands of Hawaii and complete power over Flames.
――― Earth Mother Goddesses - Potnia Theron The Authority of the Earth Mother Goddess. The power of bringing death as well as giving life. BB doesn't do anything with this other then to create life (possibly). She could, but, literally again, does not do much with this outside of keeping herself incapable of dying. This is specific to MOTHER NATURE, additionally with this skill she can speak to animals and plants.
――― Tiamat Currently in Nasu Lore, Goddess of Çatalhöyük and Tiamat are kind of the same thing. Authority to create monsters/talk/control.
――― ME PERSONALLY I do not remember all the relations each god has in general, so I might be forgetting if she would remember someone/thing she would see in the RP group: panopte. At this time, any other authorities by the other Gods are not known.

✽ General Abilities
――― Earth History She is an AI and has data of her Earth of about 4,000 years worth. Right when Sefar came and destroyed the Gods, and the Moon Cell - the data had to reset collecting. It's about that much more or less.
――― Control over the Moon Cell BB and the Moon Cell are currently one, I will say she has control over it but she is not technically it's ruler. She is more the Admin for the time being, there is a distinct difference. Honestly, this just means she has access to data from human history about literally anything - she can conceive, create, or make anything that has been created up to the year 201X.
⇥ High Capacity to process data really fast.
⇥ Knowledge of over 4000 years old?/1/?!/
⇥ A LOT OF SHIT
⇥ IF YOU PUT HER MOON IN THE GAME SHE CAN SPY ON EVERYONE DONT DO THAT
⇥ Classify her with super human genius.
――― High Level AI She can basically hack into anything, it might take her a while but nothing is too hard for her to break into.
――― Computer Smart Highly.
――― Item Creation If BB knows it exists, she can create it. She has the mind of an engineer, BB can create anything because she knows how to create it - SO LONG AS HUMANITY MADE IT then BB can too. If she cannot, then that means it is of Alien Origin and she will have to study it hard.