A cusp form is a modular form which vanishes at the cusps.
As for visualization, well... below, you'll find a rendering of the real part of the simplest cusp form of all, the modular discriminant $\Delta(q)$. Beautiful, but singularly unhelpful as far as intuition goes.
And as for the question "why are they important in number theory?", it is a very broad question. You'll have to do some reading yourself. Similar questions have been asked here before, such as this one.
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