Finally, a post I can help with.

I work for a forklift/ battery dealer.
The others are spot on. While you can run the unit off of 6 or 12 volt batteries, they won't last long and can be expensive. The problem doing that is the unit was designed with a specific size of battery in mind. If you read the data plate, you'll see it specifies a minimum battery weight, usually at least 2,000 pounds. That's because, as the other poster said, it uses the battery as a counterweight to get the necessary lifting capacity in as small of a body as possible. If you try to run it without the proper battery, and actually lift stuff with it, you may well tip it over.
If you can get the battery out of the unit, you can check the basics on it yourself. First, take your voltmeter leads and stick them into the battery cables. Operate the forklift and see how much the battery drops. On top of the battery you will see bars connecting the cells together. Take a voltmeter and check across each individual cell. You should have about 2.0 volts per cell. 2 volts per cell multiplied by the number of cells gives you the operating voltage, usually 24 or 26 volts in that type lift.
If you check the battery cells and find one or two that are drastically different then the others, you may be able to have the battery repaired by having just those cells replaced. However, depending on the age of the battery it may not be financially wise to do so. Over time, the lead plates in the battery shed lead particles down into the cell, causing positive plate growth. Once the plates shed enough, the cell will short out internally. You may replace two cells, then a month later have two or three more fail, then not long after have a few more fail. The plates usually shed at about the same rate. With 12 or 18 cells in a battery, you can spend more fixing one than a new one would cost.
How are you going to charge it? Most chargers are set up to run on 3 phase power, but single phase units are out there. Make sure your charger is set for your input voltage, too. They can be set up to operate on 208, 240, or 480 volts AC input. If you open the front door of the charger, there should be a sticker with a diagram explaining what jumpers need to go where to set it. Depending on how old it is, it may not be changeable.
Check the total voltage of your battery by sticking the voltmeter leads in your battery cables. Next, find a way to get the battery out of the unit. Be advised, it probably weighs about 2,000 pounds. Between the lead bars on top of the battery, you should see either plastic caps, or plastic caps connected by hoses. Remove the caps and fill each cell with water until it's about even with the plate you can see in the cell. Don't fill it to the top, it will boil over when you charge it. Ordinarily I'd recommend not watering the battery before charging it, but I imagine this one is probably low on water to begin with. After you put some water in it, plug it up to your charger and set it to "Equalize Charge", sometimes called "Weekly Charge". That causes the charger to run a few hours longer. The cells in the battery charge at slightly different rates, and by running the charger a little extra it brings them all up closer to 100%. That alone may bring the battery back up to a useful level, for limited use.
Hope that helps, holler if you have any other questions. What brand of forklift and battery is it?