The basic strategy is to use a key derivation function to "hash" the password with some salt. The salt and the hash result are stored in the database. When a user inputs a password, the salt and their input are hashed in the same way and compared to the stored value. If they match, the user is authenticated.
The devil is in the details. First, a lot depends on the hash algorithm that is chosen. A key derivation algorithm like PBKDF2, based on a hash-based message authentication code, makes it "computationally infeasible" to find an input (in this case, a password) that will produce a given output (what an attacker has found in the database).
A pre-computed dictionary attack uses pre-computed index, or dictionary, from hash outputs to passwords. Hashing is slow (or it's supposed to be, anyway), so the attacker hashes all of the likely passwords once, and stores the result indexed in such a way that given a hash, he can lookup a corresponding password. This is a classic tradeoff of space for time. Since password lists can be huge, there are ways to tune the tradeoff (like rainbow tables), so that an attacker can give up a little speed to save a lot of space.
Pre-computation attacks are thwarted by using "cryptographic salt". This is some data that is hashed with the password. It doesn't need to be a secret, it just needs to be unpredictable for a given password. For each value of salt, an attacker would need a new dictionary. If you use one byte of salt, an attacker needs 256 copies of their dictionary, each generated with a different salt. First, he'd use the salt to lookup the correct dictionary, then he'd use the hash output to look up a usable password. But what if you add 4 bytes? Now he needs 4 billion copies of the the dictionary. By using a large enough salt, a dictionary attack is precluded. In practice, 8 to 16 bytes of data from a cryptographic quality random number generator makes a good salt.
With pre-computation off the table, an attacker has compute the hash on each attempt. How long it takes to find a password now depends entirely on how long it takes to hash a candidate. This time is increased by iteration of the hash function. The number iterations is generally a parameter of the key derivation function; today, a lot of mobile devices use 10,000 to 20,000 iterations, while a server might use 100,000 or more. (The bcrypt algorithm uses the term "cost factor", which is a logarithmic measure of the time required.)