FTP is still used, depending on the host.
However plain, unencrypted FTP transfers are unusual nowadays. More often, FTPS (FTP over SSL) is used. FTPS is still essentially FTP for practical purposes but runs with 2048 bit Transport Layer Security (TLS), much like HTTP vs. HTTPS. Note that FTPS and SFTP are two different things.
Regarding web hosting, the only other file transfer method I've seen used regularly (besides what's been discussed) is WebDAV.
That said, if you are talking about renting servers or certain services that offer some variation of web applications, you might have FTP over SSH where FTP is tunneled via SSH (which is not the same as SFTP or FTPS). Otherwise, as @davidgo mentions, SCP (Secure Copy) is another method of transfering files.