What algorithm could/should be used to encrypt files for storage on untrusted servers? I need an encryption algorithm that works fast on large files and doesn't increase the file size. It should use a key to encrypt/decrypt data. The files will be sent using REST and over HTTPS, but a good encryption/decryption method with a key is mandatory (to be used in a website).

The files will be saved on a 3rd party servers, and although the files are sent through HTTPS, they must be stored encrypted on the remote servers and decrypted when retrieved. The key will not be distributed. The key will be stored on my trusted server, as it is also the receiver! I want to encrypt the file, put it on the 3rd party server, then when I need it I can retrieve it and decrypt it.
The question still lacks detail, so this answer will be equally vague and hand-waving.

Any modern cipher exceeds consumer disk/network IO speeds by a significant margin, you can pair any modern block cipher (AES is probably the best choice here) with a streaming mode of operation such that the file size is left intact, such as CTR or CFB. However, you still need integrity on top, so you also want a fast and strong hash function to use as a HMAC (or use AES in an authentication-capable mode, but such modes are not very widespread yet). A good hash function would be SHA256 (reasonably fast).

There is no "best" algorithm, there are only those that meet your needs and those that don't. Your needs are security and speed - choose accordingly. I gave some fairly generic suggestions above.

Of course, the better choice would still be to use something that's already production-ready, such as OpenSSL. This way you don't need to implement it yourself and just have to learn the API and make a few function calls at either end of your network sockets, and you don't have to worry about security problems (or at least, not as much).

Again, you should not be implementing cryptography if you don't know what you are doing, so your comments such as "I wish I could know that by my self" are worrying to say the least...