Without getting too deep into the definition, a trope is basically a writing cliche. It’s a story element, character type, or bit of prose that crops up fairly often, such as a chivalrous knight saving a beautiful princess from a horrible dragon. Tropes are often played straight, subverted, deconstructed, or lampshaded. Playing straight would be to adhere to the trope, subverting would be to avoid the trope, deconstructing would be to supply a logical reason for the existence of the trope, and lampshading would be to actually make fun of the trope. On it’s surface, using tropes in your writing might come across as a negative thing, but that’s not entirely the case. Some tropes can be found in every single piece of fiction ever written, or very close to it, such as the dramatic structure of introduction->rising action->climax->falling action. Not many works of fiction avoid this standard. But there are many, many tropes that are so specific that they may only apply to a very small number of works. And virtually every bit of fiction uses at least a few tropes in one way or another.
There are times when I’ve specifically avoided certain tropes, or tried to subvert or twist them to see what kind of results I get. There are other times where I’ve not even thought about tropes at all. While I don’t think it’s important to try to avoid as many tropes as possible, I do think it’s important to be aware of these tropes. What character types are played out to the point of being bland and boring? What story elements are so worn out that people get annoyed when they see them again? Not all tropes are bad, but some of them definitely are. By knowing the tropes, and specifically knowing which have been used so often that people don’t like seeing them anymore, you can improve your own work and release something with it’s own voice and its own ideas.
A really great resource for studying tropes and what works use them is TV Tropes, a Wiki-type website that collects lists of tropes for nearly every piece of fiction you can imagine. And not only is it great for figuring out tropes and finding some that you might like to avoid, it also can provide a much deeper understanding of a particular work digging into the ideas, characters, and plots that make up that work.