Martha,
The biggest determining factor is physical size. The battery compartment will only work with a battery of the proper physical size. If it is too large, it won't fit. If it is too small it will bounce around possibly damaging itself or the trailer.
There are many different battery technologies that will work. Actually, all of them will work, but some better than others.
Here are the criteria to consider when choosing a battery for your Teardrop. I have put them in order of importance (in my opinion):
1. Physical size: see above.
2. Deep cycle: allows you to take more power out of the battery without damaging the battery.
3. Cost: I put this here because everything after this point will affect the cost significantly.
4. Amp Hours (AH): this is the useable power or run time. Your original battery had about 32 Amp hours. Use that as a point of reference.
5. Life: how many years can you expect before having to replace.
6. Technology: I put this last, but technology is actually what ultimately determines Cost, Amp Hours, and Life.
Each type has it's pro's and con's.
Standard Lead/acid:
Pro: Inexpensive, readily available.
Con: Lower AH for size, shorter Life, fewer recharge cycles, liquid acid.
Gell Cell or AGM (also UPS batteries)
Pro: Higher AH, Longer life, more recharge cycles, no "free" acid.
Con: More expensive, find at battery stores only.
Note: the "Liquid acid" is only an issue when you are handling the battery. If you tip it over, or drop it, the liquid may spill. Gell cells, and AGMs have acid, but it is trapped in the "Gell" or "Absorbed Glass Mat" so it can't spill.
One of the largest factors in getting longer life out of any technology battery, is how you store it when you aren't using it. You should recharge the battery fully as soon as possible after discharging it. One or two days isn't so bad, but if you store it for a month without fully charging it, you will take a year or more off it's life.
So, always charge your battery as soon as you get home from a trip. During winter months when your trailer is not being used, you should plug in the charger for 48 hours at least once a month.
Hope this helps make your decision easier...
Craig (hijacking Betsey's account)