1st cutting is fine. That refers to how many times they mow down the field and make hay out of the grass. The first cutting usually has more weeds and is a bit stemmier (rougher, coarser texture) than others. It should be fine to give 1st cutting.
There are a number of grass hays like others have said. Timothy is the most common. Others are orchard, oat, bermuda, meadow, etc. As long as it is grass hay and not legume hay, like alfalfa as Mike said, it should be fine. Even a mix that is less than 50% alfalfa should be ok for adult bunnies, unless you find them picking out and only eating the alfalfa. I give my guys some alfalfa as a treat sometimes.