I move my hand at the wrist, kind of like my hand is twitching. Not sure if that's the right way, or if there's even a right way, but it works for me.
There are basically 2 different ways to do vibrato.

1) Moving the string parallel to the neck (side to side). This gives a weaker, more subtle vibrato, but some people like it because it vibrates the note flat and sharp. Aka the "Classical" vibrato.

2) Moving the string perpendicular to the neck (up and down). This is a more noticable vibrato but it only can vibrate sharp, so some people dont like it.

There are other methods too but those are the most widely used.

Vibrato is something that people can't really help you do aside from telling you what it is and how to do it. Listen to guitarists with good vibrato, look at a video of Steve Vai playing a song (For the Love of God and Whispering a Prayer come to mind), see how they do it, really listen to how they do the vibrato (is it soft? sudden? wild? smooth?) and try to replicate it, not on the dot but try and use it as a vehicle to improve your own vibrato.
Breaking stereotypes by playing indie on a metal guitar.

Current Gear
- Epiphone Les Paul Standard (Plus Top)
- Crappy Strat Copy (Redecorated, looks snazzy)
- Ibanez Acoustic/Electric Guitar
- Ibanez RG1570 Mirage Blue
- Peavey Vypyr 30 Watt
Try to loosten the pressure you are using with your finger tip on the string so that it is easier to roll that finger tip. On guitars with jumbo or extra jumbo frets, you can change the finger pressure while fretting. With the usual medium or standard frets on an acoustic, this will limit the range of pitch change.

Heavy guage strings, like often used on acoustic guitars, make this more difficult to do. But going to a lighter guage string on an acoustic will usually not work well, because you may not be able to get the intonation correct. In other words, the guitar will not stay in tune as you fret higher up the neck.
Some people vibrato like they bend, up and down. Others do it side to side. You won't get a good vibrato right away, so just try to vibrato really fast, until the string fades out, preferably the high e string with your middle finger. Just build some strength and after a while try different fingers and different speeds, then strings. Give it time.
practise bending slightly and returning to pitch in time with a metronome - make sure you always bend to the same pitch (maybe 1/4 tone), and always return to pitch, or it'll sound terrible. as for hand/wrist tecnique...there's lessons and columns here on UG, or youtube it.