Say you want to factorise a large integer $N$. We know (inefficient) classical algorithms to do this, a naive example being: just check all combinations of smaller numbers until you find one that multiplies to $N$. You can make this into a quantum algorithm by simply converting each operation in your classical algorithm into a reversible one (there are standard ways to do this). This leaves you with an (inefficient) quantum algorithm that solves the problem deterministically.
Here, deterministic means that the output of the algorithm is 1) fully determined by its input and 2) gives you the answer directly. For example, in the factoring case, the output of the device might be a series of qubits. You know the output by measuring these qubits and thus getting a sequence of bits, which you can put together to know your answer (exactly like what you do in the classical case).
This is clearly, however, not a very useful quantum algorithm (you might as well just use a classical computer instead). So one can try something different, e.g. Shor's algorithm. Now, this is efficient, but not deterministic. This means that running the algorithm just once will not, in general, be enough to solve the problem. To simplify, you can imagine that each run of the algorithm will give you a different output, but that having enough of these outputs you can put them together to get your answer (with the whole process being still more efficient than the classical solution).
An example of a less artificial deterministic quantum algorithm would be Deutsch-Jozsa.
Can you use probability amplification in an exact algorithm? If so, are they really all that different?
What would be the point? If the algorithm is exact/deterministic, then there is nothing to amplify. If anything, you might want to use amplitude amplification with a non-exact algorithm, to amplify the success probability. Bear in mind however that even just running non-exact algorithms enough times is usually enough to get the right answer with probability close to one.