In theory, an algorithm is said to be efficient if its worst-case running time is bounded by a polynomial in it's input length. The reasoning being that polynomials have nice closure properties. Adding, multiplying, composing polynomials are operations that yield polynomials and these are good if you reducing problems to one another.

Of course the gap between polynomial and exponential gets very very big as the input length increases so polynomial-time algorithms are way better. In practice, a polytime algorithm may take a long time before termination but it might be the case that it's an optimal (the best possible) in which case I'll say it's efficient.

In theory, an algorithm is said to be efficient if its worst-case running time is bounded by a polynomial in it's input length. The reasoning being that polynomials have nice closure properties. Adding, multiplying, composing polynomials are operations that yield polynomials and these are good if you are reducing problems to one another.

Of course the gap between polynomial and exponential gets very very big as the input length increases so polynomial-time algorithms are way better. In practice, a polynomial-time algorithm may take a long time before termination but it might be the case that it's an optimal algorithm (the best possible) in which case I would say it's efficient.