Resteeming is sharing a post by another Steemian in your blog. Those who follow your posts will also see what you have resteemed. This feature serves many useful, beneficial, and downright fun purposes. If you resteem something that later gets a lot of upvotes, you get a small curator reward bonus. If you share posts that other people might find worthwhile, you help the community make new connections and grow. And when people you follow resteem something good, you meet new people and learn new things!
However, I would like to offer some tips to keep in mind.
First, a suggestion: Don't resteem more than you post yourself. More than a 50/50 mix of resteems to original content may make it more difficult for others to see what you create yourself, and this reduces the odds that you will find community support for your own material. This is by no means a hard and fast rule, but it is definitely a good guideline for beginners who are getting a feel for Steemit.
Second, you don't need to ask for permission to resteem something. In fact, asking to resteem something is seen as spammy behavior. You should definitely upvote anything you resteem though, and consider leaving a substantive, constructive comment.
Third, don't beg for resteems. Begging for upvotes and followers is not cool, and begging for resteems wouldn't be cool either. I haven't really seen much of this yet, but it seems like something that should be said anyway.
Fourth, remember that people who ask you to pay them for resteeming your posts are probably scam artists. Track your account on steemdb.com to find out who resteems your posts, though, and you might find new people who share your interests!