Apprentice
As a Telvanni apprentice your life probably isn't easy. Of course this depends highly on your teacher, but you have to be a very lucky mer to receive a fair treatment. But well, you are an apprentice of Great House Telvanni and can consider yourself in a good position. One day you will be a high wizard yourself and you'll be able to harrass your own apprentices!
What am I doing all day?
When not busy learning or mistreated by your Masters, you are free to roam around and do as you please. Perhaps you have some kind of "homework" or an ongoing project you can bring into play. If your Master is one of the player characters you can as well have actual apprentice gameplay.
My powerlevel
... should be a fair Medium, even Low if you are at the beginning of your studies. You are capable of some wizardry, casting spells and channel magicka, but you are by no means a Telvanni Wizard. Considering this you can of course choose between an apprentice who is at the very beginning of his/her apprenticeship and a more advanced student of magic.
Where is my place in the Great House Telvanni?
By chance you are a Telvanni noble, which would make you somewhat higher ranked, but you can as well be a somebody who made it into the Great House by Oathmership or by convincing a wizard to take your character as his/her apprentice. Being an apprentice without being a Telvanni noble is also a respected position but you should be careful around nobles nontheless.
My agenda
You want to become a Telvanni Wizard. That's your agenda. If you want to achieve something else, you shouldn't mention it...
Additional information
Playing an apprentice you instantly placed your characters between the higher and the lower ranked world of the Great House Telvanni. Your character has contact with both, but you should keep in mind that there is a chance that none will like him/her much by default. Of course you can fix that by roleplaying him/her accordingly and bond with other characters. You also should have a basic understanding of magic lore in the world of the Elder Scrolls as well as knowledge about the Telvanni.