Depends upon your yard and your preferences. Gas powered mowers are most common. but you can get electric (plug in) and battery powered (rechargeable). The last type is a reel mover that has no engine at all and you supply the power by pushing it.
If you have a small yard the walk behinds (you push) are ok, if you have a big yard you might want a self-propelled (it pulls you) mower. If your property has a hill to mow the self propelled one is better, unless you are looking at mowing for its exercise value (remember the reel type)!
In my mind the engine should be an overhead cam type as they are a step up from the conventional engine. I prefer the mower to have large wheels for easy handling and the ability to mulch your cuttings eliminates the need to haul the clippings away or compost them.
As far as brands go, stick with the major manufacturers as the cheaper mowers are probably not going to last. You typically get what you pay for. I've had Honda mowers, Cub Cadet movers and an MTD mower and they all held up well.
You need to keep the machines maintenanced every year; change the oil, clean the filters, sharpen the blades. Its best to turn off the fuel and run them dry before storing over the winter. Also add a fuel stabilizer to the gas tank to prevent the fuel from going bad over the winter.