Exponential time algorithms
There's no easy answer for this one, though there are signs to look out for. Examining every possible subset of a set, for instance, is exponential - so if I had a set of integers $\{x_1,...,x_n\}$, and wanted to check every subset of these to see if they sum to $0$, I'd have to consider exactly $2^n$ subsets, which makes this method exponential time. Several different traps can make an algorithm exponential time, however, so rather than looking out for broad categories, analyse algorithms on a case-by-case basis. 
If an algorithm takes $n^2$ steps to complete, then it's polynomial. If it takes $2^n$ steps, it's exponential. The difference is the position of the $n$. If something is $O(n^m)$ for $n&gt; 1$, $m&gt;0$, then it's polynomial in $n$ for fixed $m$, but exponential in $m$ for fixed $n$.