Informally, the main difference between nondeterministic algorithms and the normal, deterministic, algorithms is that when provided with multiple choices to take, the deterministic solution will have to check one of them at a time in sequence while the nondeterministic version can cheat a bit. There are two main ways to look at it that I know of:
When faced with multiple possibilities, the nondeterministic version runs all of them in parallel, suceeding if any of the threads succeed and failing if all of them fail.
When faced with multiple possibilities, the nondeterministic version always luckily chooses a possibility that succeeds and fails otherwise.
There is also a popular definition of NP that turns out to be equivalent if you think about it:
A problem is in NP iff it is always possible to provide a polynomially long proof certificate for any positive solution to a problem instance.
For a concrete example, let's take the problem of determining if a number $N$ is composite (not prime). A way to solve this is to test all numbers less then N to check whether they are divisors of N.
If you do the tests sequentially this naive algorithm would take exponential time (since $N$ is exponentially large compared to the number of digits it has), but if you are allowed to do a magical nondeterministic choice the algorithm takes polynomial time (since each different divisor test takes polynomial time). Thus, testing if a number is composite is in NP.
And in the alternate formulation, providing a divisor of $N$ gives a polynomially long proof of compositeness.