First let's very precisely look at a tweakless blockcipher to fully understand it:
A regular blockcipher $E_k(x)$ with blocksize $n$ and key size $k$ is a permutation of the input block.
What do I mean with that? Let's first tackle the word permutation here. Often a permutation means re-arranging elements within a set. So the set of all permutations of the set $\{0, 1, 2\}$ is:
$$\{\{0, 1, 2\},
\{0, 2, 1\},
\{1, 0, 2\},
\{1, 2, 0\},
\{2, 0, 1\},
\{2, 1, 0\}\}$$
In maths, a permutation function maps the input domain to the output domain by rearranging the input domain (note that this means that the input domain must be equal to the output domain). So for example if we have permutation $f : \{0, 1, 2\} \rightarrow \{0, 1, 2\}$, then that function will map every element of the input domain to an element of the output domain, by rearranging. So if we'd re-arrange the input domain to $\{2, 1, 0\}$, then $f(0) = 2, f(1) = 1, f(2) = 0$.
This means that function $f$ is entirely decided by which rearrangement you choose. This is where the key comes in, it determines what rearrangement is used.
To summarize, $E_k(x) : \{0,1\}^k\times\{0,1\}^n\rightarrow\{0,1\}^n$ takes a $n$-bit input string and a $k$ bit key, chooses a rearrangement of the set of all $n$-bit strings ($\{0,1\}^n$) determined by the key and looks up the input string in this rearranged set.
Please note that this is all quite abstract and that the sets we're talking about are huge. This means that in a real-world algorithm no sets are rearranged or even stored, they're just mental tools to describe what happens. As an example of how this could work, define $f(x) = 2-x$ and compare with the example above.
Now we understand a blockcipher knowing what a tweak does shouldn't be too hard. It functions namely exactly the same as the key: it determines the permutation used. You might now think: what is the difference between the key and the tweak? The answer lies in something we haven't covered yet: security.
Apart from the mathematical definition we saw above for a blockcipher, there is also a security aspect. For a $n$-bit input block there are $(2^n)!$ permutations possible, and obviously a $k$-bit key can not possibly choose every possible permutation. A secure blockcipher however is defined to be a function indistinguishable from a random oracle within some pre-defined work limit (the security parameter) if the input block may be chosen by an adversary (note that the key may not be chosen).
A secure tweakable blockcipher is defined to be a function indistinguishable from a random oracle within some pre-defined work limit if the input block and tweak may be chosen by an adversary (the tweak is also passed to the oracle, obviously).
In layman's terms, the key is secret - the tweak is not.
Actual applications for such a function are immense. Much more effecient modes of operation for encryption and authentication can be made with it (OCB3), hash constructions (BLAKE2, Skein), efficient full-disk encryption, etc.
The most important reason why it's useful is that you can overcome the limitations of the electronic code book. By using the tweak as a little counter you can make every single invocation of the blockcipher essentially an unique function unrelated to any previous calls. This is very comforting for security.