One feature that the PS Vita has carried over from the PSP is Remote Play. What Remote Play does is allow you to access the content of your PlayStation 3 from your handheld, using a wi-fi connection. Remote Play on the PSP never really became a big deal, partly because the lower specs and lack of second analog stick meant there was only a limited number of things you could use it for. It's hard to say so soon how important Remote Play will be for the PS Vita, but the better specs of the system and its second analog stick should make it at least a little bit more useful.
PS Vita-PS3 Pairing
The first thing you need to do to enable Remote Play (and I'm assuming here that you have both a PS Vita and a PS3), is pair your devices. It's pretty easy to do, as long as the PS Vita and PS3 are fairly close together (like, in the same room).
First, turn them both on. Go to the "Settings" menu on the PS3, select "Remote Play Settings", then "Register Device", and finally "PS Vita System." A number should appear on your PlayStation 3 screen. Do not select "OK" yet. Then, on the PS Vita, select "Remote Play", then "Start", then "Next." You should then see a place to enter the number your PS3 gave you. Enter the number and select "Register." If all goes well, you'll get a message to tell you the process has been successful. Finally, select "OK" on your PS3.
If you need to change paired devices, the process is pretty much the same, except when you select "Remote Play" on the PS Vita, you'll need to ignore the connection options and select "Options" then "Settings" then "Change Connected PS3 System."
What You Can and Can't Do Via Remote Play
It would be really cool if you could do everything your PS3 is capable of remotely on your PS Vita, but sadly, you can't. Some of the restrictions make sense, while others are kind of silly. You can get at your PS3's Settings, Photo, Music, Video, Game, Network, PlayStation Network, and Friends menus (I'm not really sure why you'd want to access the PSN or your Friends via the PS3 from a PS Via, when you can do it directly on the PS Vita and get the same information, but there you go).
What you can't do is use every single feature on those menus. The Settings, Photo, Game and PSN menus will only let you access some features. In addition, you won't be able to play all your PS3 games. The ability to use Remote Play to play PS3 games has to be built into the game, so whether or not this happens in future games will probably depend on how much people use the feature. And that will probably depend on how many games have Remote Play enabled. Yes, it's circular. The bottom line is, if you want to play PS3 games on your PS Vita, use it a lot right off the bat, and be excited about it online so it'll keep being included.
Finally, and this seems like a really silly restriction to me (but maybe it's simply a hardware issue?), not all videos on your PS3 will be available to watch on your PS Vita via Remote Play. You won't be able to watch any discs, either Blu-Ray or DVD, and any copyright-protected files (since pretty much all content is copyrighted, I'm guessing this means files with DRM, but I could be wrong) will also be off-limits.
Most of the controls for Remote Play are as simple as using the equivalent buttons on the PS Vita to navigate the PS3's menus. A few exceptions, like the PS3's PS button and changing image quality or screen modes, require tapping the PS Vita's screen and selecting the operation you want to perform.
Three Ways to Connect
To use Remote Play once you've paired your devices, all you need is wi-fi. If you have a PS3 with built-in wi-fi network capability (more recent models, in other words), you just select "Remote Play" then "Start" on the PS Via, and "Network" then "Remote Play" on the PS3. Finally, select "Connect via Private Network" on the PS Vita and the two machines will establish a connection. The advantage of this method is that you don't need anything other than a PS3 and a PS Vita to play. The drawback is that you have to keep the PS Vita within range of the PS3's wi-fi.
If your PS3 is a model that doesn't have built in wi-fi network capabilities, you can connect via your home wi-fi network. All PS3s come equipped to connect to a wireless home network, and so do all PS Vitas. Follow the exact same steps as using the PS3's built-in network above to connect the devices. Here the advantage is you can use any model of PS3, and the disadvantage is that you can't connect this way without a wireless router. You also need to keep your PS Vita within range of your router.
Finally, if you want to be able to get at your PS3's content when you're out and about, you can do that through any available wi-fi. Your PS3 needs to be connected to the internet, but it can be a wired or wireless connection (so if you're still running cables everywhere, you can use this method even if you can't use the above two). Connecting is much the same as if you were at home, except you select "Connect via Internet" on the PS Vita (instead of "Connect via Private Network." The drawbacks of connecting this way are that not all wi-fi networks will let you do it, and you have to put the PS3 into Remote Play mode before you leave your house, as there's no way to do it remotely.
When you're done, turning off Remote Play is as simple as switching to another application on your PS Vita. The connection to your PS3 will automatically close after 30 seconds (the PS3 will remain on and in Remote Play mode, however). If you want to also turn off your PS3 remotely, first tap the PS Vita screen while still in Remote Play and select "Power Off". The PS3 will shut itself off and the connection will close.