Courtiers are unlanded characters.
Your courtiers can serve in many of the same positions as landed direct vassals: as council members, flank commanders, and holders of honorary titles. You have full control over their marriages and education.
Characters residing in your court include most of your courtiers, along with some other characters temporarily living in your capital while retaining their existing liege. The distinction between your courtiers and characters residing in your court is discussed in the final section of this article.
Gathering courtiers
You can often gather the courtiers you want through invitation, especially if you have high state diplomacy and likable traits. If a courtier will not accept immediately, calculate whether you can clear the necessary opinion difference for accepting invitations using gifts, seduction, or letting them educate up to two minor children or grandchildren. (In the character search window, characters that can be swayed by a gift will have a yellow horizontal hand instead of a red thumbs down.) With the 
Conclave DLC, enough wealth and the right conditions, consider buying a favor from the character and using it to force the invite to your court. This is the only way you can invite characters of another religion over to your court.
If they can marry, request to have them marry into your court.
For skills
Talented councillors: In the character search tool, select adults, not imprisoned, not rulers, and men (if gender-restricted), then sort by skill. Start with a chancellor, whose diplomacy will help you invite others.
Champion commanders: If you are headed to war, you may want commanders who specialize in army movement speed, siege speed, or combat. In addition to sorting by martial skill, consider searching by trait names such as "patient" or "organizer".
Guardians: Having educators in your court lets you choose tutors for all your courtiers, and often for courtiers of vassals (such as their heirs). While you can get your own children and grandchildren educated almost anywhere, having the guardian in your court ensures they will not change the child's culture.
If you're a very patient player, consider inviting parents of highly skilled children and getting them educated according to their strengths.
Catholics may want to identify strong cardinal candidates or taxation-maximizing antipopes. Followers of Indian religions can steal other rulers' Court Gurus by searching for "mastermind scholar patient". With 
the Reaper's Due DLC, consider stealing courtiers with very high Learning to become your Court Physician.
For power
Claimants: Press a claim for someone in a way that will make them your vassal, or arrange to inherit a claim for your dynasty's next generation. Claimants are generally willing to accept invitations if their current liege holds or controls the claimed title.
Heirs: Make your dynasty powerful by getting male heirs into matrilineal marriages with your daughters, and vice versa. Arrange for foreign duchies to inherit into your kingdom. Also consider pretenders, heirs of heirs, and anyone else who might inherit a title after a few murders.
Potential spouses and concubines: For whatever your goals are: quickly expanding your dynasty, eugenic breeding, gaining alliances and claims, or simply gaining prestige. See also Advanced marriage guide.
For money
You can inherit money from rich courtiers in two ways. First, if they have no parents and no living progeny, you will simply inherit their gold and artifacts when they die in your court. Many lowborn characters, especially those with high skill, will earn over 20g during their time as council members. Gifts you send to these characters are great investments.
Second, if they have no valid heirs, you can grant them a county with the intent of making them lifetime administrators. Upon their death, you will regain the titles, along with any gold, retinues, and tech points. You can assassinate them to speed up the process. Or you can appoint an heirless old man with a middle-age wife, and let him die naturally. This is a good way to deal with counties in excess of your demesne limit (and duchies in excess of two) that you might want later.
You can also take money from rich prisoners by banishing them. If you have a valid reason to imprison someone, such as a known plot or excommunication, you can imprison and banish them once they are your courtier. If they are already your prisoner, but answer to another liege, you can invite their spouse and then banish them.
Court size
Your court may grow through invitation, procreation, decisions that generate courtiers, and events. Banished characters and ousted rulers may also land in your court. If a courtier's marriage ends, they will often(?) move to the court where their parents live.
Your court will also grow or shrink due to marriage, because spouses move together when they can. A courtier who marries a ruler will move to the ruler's court. When two courtiers marry, the wife moves to the husband's location, or vice versa for matrilineal marriages.
Once you hit 30* people in your court, your courtiers receive reduced fertility. If you need a couple to produce multiple children, you may want to consider landing them. Alternatively, you could marry off some unneeded single adults to reduce the size of your court. (*Some mods may change COURT_SIZE_CHILD_PENALTY_THRESHOLD.) Additionally, a court with 25 or more courtiers is more likely to spread disease.[1]
AI courts with 10 or more courtiers[2] will be periodically "pruned" of "useless" characters in order to prevent the game from slowing to a crawl as time advances.. Characters under 40,[3] with health at least 7,[4] those with claims, and certain others (e.g. who caused someone's death, and immortal characters in mods) are immune to pruning.
Make sure to keep most of your courtiers happy - high-intrigue courtiers who dislike you can bring a lot of plot power to assassination plots against your ruler and family.
The player's court is not subject to pruning (the outright killing of courtiers in AI court), but unimportant characters still receive important penalty to health and fertility. The fertility decreases the number of children that characters can have by one. The health penalty does not affect anyone who has a living parent, nor anyone with a living spouse, a job, or an honorary title. It does not apply to children. It does not apply to character's of the court owner's dynasty.
Characters residing in your court
Your courtiers usually reside in your court, but so do several other types of characters. Characters also reside in your court if they are your prisoners, landed council members, or children being educated by adults residing in your court. You, too, usually reside in your court.
Characters residing in your court form their opinion of you based on your Personal Diplomacy attribute, instead of your State Diplomacy total. Residence location is also relevant for assassination plots and capture in sieges. Your courtiers who are not residing in your court are considered "abroad" and are listed in a separate tab of your character screen.
References
Meneth's explanation on pruning