Essentially, a cut-free proof is a proof that does not use a lemma. That is:

Given $$(A\wedge B\wedge ...)\vdash C$$, and $$C\vdash (D\vee E\vee ...)$$ we can cut $$C$$ from the proof of $$(D\vee E\vee ...)$$ when $$(A\wedge B\wedge ...)$$ is true. Here we consider $$C$$ to be the lemma in the proof of $$(D\vee E\vee ...)$$.

The ability to eliminate $$C$$ in the above is called the cut-elimination theorem. 

For more information set [[http://en.wikipedia.org/wiki/Cut-elimination Cut-elimination]]