For medium to thick fitness flooring (anything thicker than 0.4" or 10mm), one of the most popular options is simple square rubber tiles. They might be fairly expensive, but they are easy to install, shock-absorbent, easy to clean, and durable.
A good number of fitness flooring tiles nowadays feature a jigsaw locking system. These might be heavy enough to be fitted without having to use adhesives, but the thinner the flooring is, the higher the possibility of sections coming loose.
The installation of flooring rolls is a lot less labor-intensive and they are often cheaper per square meter than tiles. Depending on the flooring's thickness, the rolls come in a variety of lengths.
Many people prefer rolls because they don't have as many joints as tiles, and they, therefore, provide a smoother appearance to the floor's surface. Fewer joints also have another benefit: the chances of gaps forming or sections coming loose after heavy usage are less.
These typically consist of large sheets of varying thicknesses and sizes. For weight-lifting exercises, a good quality 1/2-inch rubber mat should suffice, but for more heavy-duty activities you will probably need at least a 3/4-inch thickness. These mats are normally sold in sets of interlocking "puzzle pieces". Beware that inferior quality materials will basically just spread apart over time.