Your in-game UserID is the login name you use to log into the game. Usually you'll see it with an @ in front of it.
When signing up on The Elder Scrolls Online website, players are asked to create a unique UserID (no two users can have the same UserID). This UserID is the name of the account that will be requested for every connection to the game and website.
It is also a publicly displayed name that will be visible in-game by friends and guild mates. This allows players with several characters on the same Megaserver to communicate with their friends no matter what character they're using. Additionally, it simplifies the in-game mail system and allows messages to be delivered to all the characters of the account.
Please note that the in-game systems of The Elder Scrolls Online require players to type a "@" symbol in front of a UserID to distinguish it from character names.
The Character Name is the name that is proper to individual characters. It is the one that is displayed on the public chat channels. Players who don't know each other well enough will have to use the character names to /whisper each other.