Hashing versus encryption
The biggest different between bcrypt and blowfish is that blowfish is a symmetric encryption algorithm and bcrypt is a hashing algorithm. (It is a special kind of hashing algorithm, but we will get to that later.)
Encryption (what Blowfish does)
A symmetric encryption algorithm takes a secret key and some plaintext (the term "message" is regularly used and the variable m is typically used to refer to it) and outputs ciphertext, c. The idea is that the ciphertext can only be transformed back into the original message with same[^same] secret key that was used when encrypting.
[^same]: I am giving approximately true answers here to focus on the central ideas and concepts. I am hoping that those who recognize where my answers aren't precisely true will cut me some slack for expository purposes.
So Alice might encrypt the message "Attack at dawn" with the key 13 (this is not blowfish, but it is symmetric encryption) and produce the ciphertext "Nggnpx ng qnja". She can send that ciphertext to Bob, and if Bob knows that the key is 13, he can transform the cipher text back into the original messages. Sometimes Alice doesn't want to send a message to another person, but she wants to store it for herself to read at a later time. We can think of those cases as Alice sending a message to future Alice.
With Blowfish (instead of the encryption that I used in my example), it is effectively impossible for anyone who doesn't have the key to learn anything about the original message (other than its length and existence) from the ciphertext. Just an aside, AES is a preferable symmetric encryption algorithm over Blowfish, but I will continue to use Blowfish in my descriptions as that is what was asked about.
A cryptographic hash algorithm (bcrypt is special kind, with some extras, but I will start with a simpler case) does not take a key and is not practically reversible. So a cryptographic hash algorithm like SHA256 would take a message (let's say a password) like "JaguarsRule!" and transform it into a hash.
$ echo -n 'JaguarsRule!' | shasum -a256
One of the several properties of a cryptographic hash is that you can't feasibly compute "JaguarsRule!" from
13f26e5a60e402d895c5ce1d9492d080563c5079a8b5f52a25953fd24a2cb1da. There is no key in this case.
So a straight forward hashing mechanism like this might seem like a good way to check passwords on a server. Suppose you are running some web service that has people log on. You have a user, let's say with username
jason who sets up his account with the password
JaguarsRule!. Because you don't want anyone who can read what is on your server to learn Jason's password, you don't store the password itself. Instead you store its hash.
When jason (or someone pretending to be jason) logs in your server takes the password that gets entered and performs the hashing operation on it and compare that hash that is stored. If the hashes match then the system knows that the correct password was entered.
Salts and slow cooking
The scheme described above, using a general cryptographic hash like SHA256, would work great except for one thing: People pick guessable passwords. If people were as likely to use passwords like
wZFoO_SKrgEw as they are to
JaguarsRule! then we wouldn't need specialized hashing functions like bcrypt. But people, being human, are not going to pick passwords randomly and uniformly over a sufficiently large space. Instead, they are far more likely to pick some possible passwords over others.
Consider another user of your system,
pillboy. Pill Boy is also a fan of the Jacksonville Jaguars and happens to have also picked
JaguarsRule! as his password. Using (unsalted) hashing, the hash for Pill Boy's password will be the same as the hash for Jason's password. An attacker only needs to compare hashes to see that those two users have the same password.
Furthermore, if an attacker discovers one way or another that Jason's password is JaguarsRule!` then she immediately knows Pill Boy's password. Furthermore, she might create a big list of hashes for the most common passwords.
The solution to this is to salt the password with something unique before hashing it. So when first creating a the hash of Jason's password (when Jason first signs up and creates the password), you'd add some random stuff to it. Suppose that at this time your system creates the salt
9cb3779f69e271c1. It then sticks that in front of the password before hashing, so it is really hashing something like
9cb3779f69e271c1JaguarsRule! (This is not how the salt actually gets added, but this is close enough to explain what salt is for.). The hash of
9cb3779f69e271c1JaguarsRule! will be completely different than the hash of
The system will need to know what the salt was when it comes time to verify a password. So it will store the salt and the hash along with the Jason's username. When Pill Boy creates his account, it will be set up with a different randomly generated salt, and so his salt and password combination will produce a different hash.
So what you store on your server might look like this
By having different salts for these users, we can't see that they have the same password, and the attacker won't be able to compare the hashes to a list that she's precomputed of hashing of common passwords.
So bcrypt, because it is designed for password hashing, takes a salt.
Taking it slow
Salting is absolutely essential, but it still doesn't stop an attacker who gets hold of what is stored on the server from automated guessing. She can take the salt and combine that with common passwords and compute hashes until she finds a match. She might be able to perform tens or hundreds of millions of password guesses per second.
One of the reasons that she can compute some many guesses so quickly is that cryptographic hash algorithms like sha256 are designed to be fast and efficient while still meeting their security properties. So things like bcrypt do are "slow hashing". The idea is so that getting from some message (salt and password) to a hash might take so much computation that a typical computer will take a quarter of a second to perform the hash.
The user won't notice a quarter of a second delay, but if the attacker is using the same kind of computer, then she ends up being reduced to making four guesses per second instead of 10 million. (In practice, she will have specially configured computing equipment, but slowing her down from millions of guesses per second to tens of thousands of guesses per second is a good thing.)
bcrypt is one of several "slow hashes". Others are PBBKDF2, scrypt, and Argon2. Those last two not only require a lot of computation to create the hash, but also require a lot of memory.
Bcrypt is great, but ...
Salting and slow hashing is something that every password hashing system should do. But it also yields diminishing returns. There is just so much that slow hashing can do for the defender, and so we still need to (as long as passwords exist) get people to use harder to guess ones. I've written more about that in Bcrypt is great, but ...
NB: Thank you https://security.stackexchange.com/users/6253/schroeder for correctly deleting my previous answer in which I took "like I'm 5" literally. And my apologies to the original poster for having been snarky instead of helpful.