How can a superposition of 0 and 1 represent any discrete information at all?
The inputs to a quantum computer are always the same as the inputs to a classical computer; a definite sequence of 0s and 1s. This superposition thing is something that happens in the middle of the computation, and what you're quantum algorithm tries to do is make sure that your output is also a definite sequence of 0s and 1s. So, you're not using the superposition to represent any information in the sense of an input or an output.
What about the logic gates?
In many ways, logic gates work just the same as in the classical case. For example, there are many gates which, given an input string of 0s and 1s output a different string of 0s and 1s. Examples include the not gate, controlled-not and Toffoli (controlled-controlled-not). For a quantum state, if you have it written out as a superposition of different strings of 0s and 1s, these gates act on each of these strings just as they would in the classical case. In this sense, you can do everything that a classical computer can (the Toffoli gate, in particular, is said to be 'universal' for classical reversible computation, i.e. arbitrary combinations have the full power of classical computation).
However, quantum computation has more logic gates. It has gates which can create and recombine superpositions, and gates which can change the complex argument on the coefficients of the superposition (the $\alpha$ and $\beta$ in a statement $\alpha|0\rangle+\beta|1\rangle$).
How can a discrete decision be taken by leveraging this superposition principle?
The trick, as mentioned above, is to make sure that there is a definite output, i.e.e that there is no (or almost no) superposition when the information is read out at the end. Where this superposition becomes useful is it gives these extra gates, over and above the classical stuff, some room to work. Without specifying exactly how they do it, you can easily imagine how having access to an additional set of abilities can sometimes be combined in new ways to give faster computations.
Combination of 0 and 1 is basically important for computing. How can this third state of qubits be leveraged to give a boost in computing?
The insight about how it works is to emphasise first that the superposition of a qubit is not only one extra state. It is an infinite number of them, because instead of either being a 0 or a 1, it can be any state $\alpha|0\rangle+\beta|1\rangle$ that satisfies the constraint $|\alpha|^2+|\beta|^2$. What's more, when you combine $N$ qubits together, you can get superpositions across all $2^N$ different sequences of 0s and 1s.
Now, how is this useful? So far, we only know that it's useful for a limited set of problems, and these problems often require knowledge of some global property of a function, i.e. I don't just want to evaluate some function $f(x)$ for a specific value of $x$, but I want to know some relation between many different values of $f(x)$. The superposition can simultaneously evaluate all the different values of $f(x)$, and then it just needs a bit of magic to work out if it's possible to recombine these superpositions in order to get out the answer we're interested in. The point is that this opens up a completely new possibility for the way an algorithm could work.
