Nikon D5100 Lens Help

Preface: I'm am a complete camera n00b. I know how to turn it on and take the picture but beyond that I am totally uneducated; I've been wanting to take a class but can't seem to find the time...
Question: I want to buy my wife a new lens for her Nikon D5100. When we first got it, it came with a 18-55mm and a 55-300mm but the 55-300mm died on us (won't auto-focus, just clicks like the motor is stuck). The 18-55mm is great for everyday photos but she wants a zoom lens for vacations and events. I don't want to spend an arm & a leg but understand that these can get spendy. So before I just go out and buy something that a salesperson recommends, I figured I'd signup here and ask you folks.
Thanks in advance for any assistance!!

Welcome to the forum. Was the 55-300 meeting your needs? Did you check to see if your 55-300 was still in warranty? Or was it dropped or damaged in a way the warranty wouldn't help.

I thought that it was meeting our needs, yes. It was not dropped or damaged as far as I could tell... it just worked one day and not the next :| I, foolishly, did not send in the warrantee card when we bought it. I'm going to check in our storage to see if we still have the box and receipts still when I get hime tonight but I'm pretty sure we threw all those boxes out earlier this year.

Yeah, that does make it hard to do then. There are alot of lenses out there. What is your price range?

That's a very good question. I'm honestly not sure where to start... I don't want to buy something that is low quality but at the same time I don't want to spend a ton either. I'm thinking under $400 if possible.

I have the 70-300 on mine and love it - a great lens for the price.
As my walk around I have the 18-105 so I don't have the gap you would have between 55 and 70 although that may not be so crucial.

Nikon has refurb. 18-140 at that price point. A very nice lenses. You can get 70-300 or a 55-300 in that range as well

Can I get a little more help? I know that the numbers represent millimeters (right?) But what do they mean? I'm assuming that the higher the number the further you can zoom?

Yes, that's what it means.
It also equates to how wide it's field of view can be.

I hate to ask but did you check to see it the switch on the side of the lens was switched to M(anuel) try flipping it back and forth a couple of times.

lol I understand. I did say i was n00b! But yes, I checked that. Thanks for all of your replies!