Encryption is twice safer than encryption once, although this may not be clear at first.
Intuitively, it seems that encryption twice with the same algorithm does not provide additional protection, because an attacker can find a key that decrypts all the way from the final cyphertext to plain text .... But this is not so.
eg. I start with plaintext A and encrypt with key K1 to get B. Then I encrypt B key K2 to get C.
Intuitively, it seems reasonable to assume that there might be a key, K3 , which I could use to encrypt A and get C directly. If so, then the brute-force attacker will eventually stumble on K3 and be able to decrypt C , as a result of which an additional encryption step is not added security.
However, it is unlikely that such a key exists (for any modern encryption scheme). (When I say "very unlikely" here, I mean what a normal person would express with the word "impossible").
Why?
Consider the keys as functions that provide display from plain text to cyphertext.
If our keys are all KL bits in length, then there are 2 ^ KL such mappings.
However, if I use 2 keys of KL bits each, this gives me (2 ^ KL) ^ 2 mappings.
Not all of them may be equivalent to one-step encryption.
Another advantage of encrypting twice if two different algorithms are used is that if a vulnerability is detected in one of the algorithms, the other algorithm still provides some security.
As others have noted, a coarse forced key is usually the last resort. An attacker often tries to disrupt the process at some other point (for example, using social engineering to detect a passphrase).
Another way to increase security is to simply use a longer key with one encryption algorithm.
... Feel free to correct my math!