I'm not sure what kind it is. Here's a picture:
http://static.flickr.com/28/60943185_6357bb59e9_o.jpg

Here is a web site that might be beneficial for reference. Rosemary is a shrub when grown in climates where it thrives. Normally if kept inside rosemary needs plenty of sunlight and a cool temperature. Good luck. EP

Here is a link that might be useful: rosemary

Here is anothe link with information for bringing in rosemary for the winter. EP

Here is a link that might be useful: rosemary in container

It should do fine on a cold window-sill but you might want to move it to a slightly larger pot. If it dries out, it will croak and it's too hard to keep a 3" pot damp but not wet.

A 3 inch pot is way too small for rosemary. It either needs to be in the open ground if your climate allows or in a bigger pot. Eventually it could need a container at least a foot across or more. If your apartment has a balcony or step I would leave it outside as long as possible. It can stand a light frost when mature. It does not like being indoors and should only come in when it really has to.

Are the leaves still ok to use fresh if we've had a hard frost?

I used to bring in my rosemary in the winter, but when they would get large they would have a tendency to become rootbound, even if in a pot larger than the upper part. They would constantly be wilted, even when watered 2 or 3 times a day, due to this. I had some live through a few mild winters, but eventually a severe one would kill it, and I would have to start over.

Now I have a huge rosemary plant, about 3'x4', and over 3' tall. The truck is about 2 1/2" in dia., and it flowers almost constantly (took about 3 years for it to start). To save it during cold spells I simply put a tarp over it, weight it down with bricks, and keep a lamp inside with a 60w bulb on at all times. Last year I left it on for 10 days when the temp was below 12º constantly, and it only suffered brown tips where it touched the tarp. I do this if it gets under 15º for any length of time, and if it is under 20º I just throw the tarp over it, and remove it during the day. It is about 10 years old, and I have more rosemary than I can ever use!

Dave

BarePaw, rosemary is evergreen where it will survive which is about zone 8 and warmer (zone 7 for the arp cultivar, or if you are covering to keep warm). It stays green and keeps it's leaves all year unless it is sick or dead.

Yes, you can use the leaves if they have frozen. But it isn't going to survive outside in zone 5.
I have five indoors - in 10-12" pots. They were repotted early this month and are now in bloom in a sunny window. The oldest is 8 years old. I use a regular potting mix (Miracle Gro or Schultz or Fafard) and repot every year. They need pinching back during the winter when the growth gets straggly and then they go outside in May.
Indoors they need as much sun as possible and must be kept watered but not soggy.

My mother brought me a rosemary topiary for Thanksgiving, and I think the first thing to do is re-pot it.

I usually don't have any problem wintering over my rosemary in the apartment, but those are very airy and open in comparision to this. I'm worried about it getting the rosemary winter crud! But it is so very pretty. If I can get it thru the winter, I'm going to plant it out in the garden at work this summer.

If rosemary winter crud is mildew, I've found a pre-mixed neem oil spray in a squeeze bottle does a fine job on it.
Usually don't have trouble with mildew indoors since my water has sulfur in it but every once in a while, things happen.
Since I'm a rather frail very senior citizen, I'm not a believer in big pots. By repotting in fall and doing a bit of root pruing, the rosemary can stay in a 12" pot which is about as big as I want to handle. Good idea to have 2" of fresh soil under and around the root ball so it doesn't dry out so fast. (Think bonsai!)

BarePaw, I'd suggest only going up to a 4" pot - I've got several that have lived multi-years in such. It takes a long time to grow to where it really "needs" a pot bigger than 6". 3" is too small - it will indeed dry out when you're not looking.
The cold windowsill should be just fine, though. My experience is that the colder you can keep it within its hardiness range, the less you have to worry about it stretching from lack of light. I have most of mine under lights now, but I need them up within 6 inches of the lights in order to balance the heat from the light fixtures (no other heat in that room).
Alison, my understanding of 'rosemary winter crud' would be not mildew, but the fungus that causes leaves in the middle of the plant to die. I've only been able to control that by opening up the center for air circulation. I haven't tried neem oil for mildew, but I've used the baking soda solution touted elsewhere on this site, with success. By all means repot (any time after mid-January), but don't count on keeping a topiary going outside in Ohio. Topiaries are more sensitive to extremes than a normally grown plant, and have less "redundancy" to survive stress. Find a good spot inside, and summer it on the patio.

I have one that I have had for many years in an 18" ceramic pot. I bring it in every fall, it is the size of a small tree. It sits between east and south facing windows and drys up between watering. I have never had any trouble with it. I always thought it made a great house plant.