As you probably heard by now BTC-e appears they may have gone the way of the GOX! After being around for years, rumors are they are gone, possibly with no intention to return deposited funds they still control!
As of writing this is their homepage: https://btc-e.com/
What is a Private Key?
Private Keys are the building blocks for how cryptocurrency works. Your Private Key(s) unlock your coins on the BlockChain! Without the correct Private Key, you can not access your funds.
What is a Wallet?
Wallets are nothing more than a piece of software that you use to access and interact with the BlockChain. Some are Command Line Interfaced more commonly known as CLI, others more common among the average user are Graphical User Interfaced or GUI. Both types of wallets simply allow you the user to do all the different things you want with the BlockChain.
How do Private Keys and Wallets work together?
Your wallet stores the entire BlockChain (or at least has some form of api access to it), but without the proper key it has no way to know how many coins you control and can not send or receive coins! You load your Private Key into your Wallet in order to access your funds and perform all the functions you expect such as Send and Receive.
What do exchanges have to do with all of this?
Exchanges do NOT give you your private key. They control the Private Keys and therefor THEY have full control over YOUR funds! When an exchange (or and 3rd party) decides to perform the classic "Exit Scam" or just simply gets hacked, YOUR funds are stolen! You have almost no recourse to get your funds back and you accept a full loss! By controlling your private keys and learning to secure them properly, you have a much smaller risk of your funds being stolen than keeping them on an exchange.
Read more
https://www.coindesk.com/coindesk-explainer-btc-e-arrest-mt-gox-connection-big-news/