Ah, the mysterious 1099 MISC form. I wish I had a nickel for every time someone asked me about that. I'm going to answer your questions in reverse order, I think it will make more sense that way.
First, non-employee compensation means that you did work for the company as an independent contractor. Maybe you did some freelance work for them before they actually hired you--that's one possibility. I had another case where a woman worked for a government agency and she received a W2 for her job. But--part of her work was funded by as special grant and they paid her for that as a 1099 employee. She now keeps really good records of which part of her job is what, but that was kind of strange. Normally, you don't get a W2 and a 1099 non-employee compensation from the same company-but it can happen.
Now if you get the 1099 MISC for non-employee compensation, that is automatically considered by the IRS to be self employment--in which case you are taxed for 15.3% for your self employment taxes PLUS tax at your regular rate.
What you can do: First, make sure the income really is yours. Since you did receive W2 income from the employer, go ahead and ask about the 1099. I've worked on a few returns where the company sending the 1099 made a mistake (it's rare, but it does happen.) Second, assuming that the income is yours and you were supposed to claim it, look to see if you have any job expenses against that income. We often have job expenses for our W2 jobs that we can never write off because there are so many other pieces to the puzzle--but, with Schedule C income (which is what you've got there) you can write off your expenses directly against the income. That would reduce the amount of extra tax you pay.
I've attached two links, one about contract labor and one about claiming business expenses.