RO = Reverse Osmosis
RO/DI = Reverse Osmosis with Deionizing Resin
HN1 already covered a bunch, but the difference between RO and RO/DI is a matter of pureness.
The use of a RO membrane will take out about 95% of the junk in your water. So if you started out with water with a TDS (total dissolved solids) reading of 100 ppm, your water coming out would be around 5 ppm. Better than what you started out with... but still not "pure".
The use of DI resin after the RO membrane will take out the rest of the junk that the RO membrane missed. So with RO/DI, regardless of what you put in, you TDS coming out should be 0.0 ppm - as close to pure as you can get.
DI resin gets depleted, so it has to be replaced from time to time. How often you need to change it will depend on how bad your water is to start with. Best way to know when it needs changing is by using a TDS meter, and measuring the TDS of the water coming out of the unit, from time to time.
Whether or not you need RO or RO/DI is a matter of choice. It all depends on how pure of water you want. My opinion... if you're going to the trouble of making RO water, add the DI stage and know you're getting as good as water as you can possibly get. Why go to the trouble of doing RO, and just get 95% pure water. Who knows what's in that 5%!