Two-slot outlets should generally be replaced with three-slot outlets, but the best way to add grounding isn't easy. The two slots in old outlets represent the hot and neutral wires. Since the mid-1960s, the electrical code has required a third slot for an equipment ground, which adds shock and fire protection. The best way to get this protection is to run that third wire, which is usually bare copper or wire with green insulation, from the outlet box back to the grounding bar in your main electrical panel. However, you usually have to open walls or floors to get the wire in—a big job. Sometimes grounding can be done in other ways, but they're often tricky and require the expertise of a licensed electrician.
A second way is to install a GFCI outlet. This device will give you better safety protection than the equipment ground, without running the third wire. However, this method has drawbacks. Many types of modern electrical equipment, such as air conditioners and computers, won't operate properly without an equipment ground, which is why you're required to label any GFCI that doesn't have one. (The labels are included with the GFCI device; see the label in the photo.)
It's unsafe to use adapters that convert three-prong plugs to two-prong ones, because as normally used, the adapters bypass the safety features of the equipment ground.