A gas grill is the most common kind of grill sold these days. It offers good flavor, versatility and convenience. Most gas grills require a fairly good amount of space and can not be used too close to any structure. If you don't have a large area then you probably can't use a gas grill. Gas grills can have side burners, rotisserie kits, multiple burners and a whole bunch of features. Gas grills require fuel either from a natural gas line or from a propane tank. The grill must be equipped for the specific fuel. The great thing about a gas grill is that you can heat it up and be cooking in as little as ten minutes.
Charcoal grills offer better flavor than gas grills and also require a good amount of space to avoid unwanted fires. Charcoal grills require that you buy charcoal and that you have plenty of time to get the fire going. Generally you should plan on 20 to 30 minutes from the time you light the charcoal until you are ready to grill. Also, unlike a gas grill, charcoal can't be simply turned off. You can, or course dowse coals in water but this can be damaging to a charcoal grill. If you grill frequently you will end up spending a lot more in fuel with a charcoal grill than you would with a gas grill. While most charcoal grills cost less than gas grills the fuel cost will more than make up for the difference.
Electric grills offer the least authentic grill flavor but can generally be used almost anywhere. Electric grills can heat up quickly and don't require any fuel other than an electrical outlet nearby. However you really do lose a lot in the way of flavor. Also, many electric grills do not generate a very high heat that is required for searing foods on a grill. If you are restricted by local laws or limited to a small balcony then you might only have an electric grill option. Otherwise, I really recommend against them.
There are other grills out there like pellet grills. A pellet grill burns hardwood pellets that are made from wood scraps. They offer a true hardwood flavor that is superior to charcoal grills. However pellets can be hard to locate in some areas and they can be expensive. Pellet grills are also not cheap, costing about as much as a good quality gas grill. Pellet grills are also good smokers so you do get that additional versatility.