Step 1 - Loosen and remove the old strings
In this picture the demonstrator is using a String Winder, rather than his hand. String winders are faster and often come with handy extras like string cutters. I'd recommend the Planet Waves Pro winder/cutter.
If your guitar has a normal Tremolo bridge, once the string is detached from the tuning peg push it gently backwards and the ball end will appear out of the plastic casing on the back of the guitar.
If it's a Tuneomatic bridge, it will look something like this -
The same method applies. Just push back gently on the string and the ball end will appear out of the back of the tail as above.
Step 2 (Optional) - Clean the Guitar
While the strings are off it's a great chance to clean places you can't normally reach like between the pickups or the fretboard. Try to avoid furniture polish (use very sparingly if that's all you have) and use Lemon Oil on the fretboard.
Step 3 - Replace the Strings
Feed the strings back in the way you took them off - reverse that process, effectively. For the Fender style Tremolo pickups, you may need to remove the plastic casing from the back of the guitar. Don't worry if that's the case!
Pull the strings back through to the neck and the tuning pegs.
My technique is to pull tight to the nut (white string guide at bottom of above picture) and hold there with a thumb or finger while I wind around the peg a few times (as above). Once you're satisfied, push what's left through the eye of the peg.
If you have vintage style slotted tuners as above or an older model, there's a different technique you'll need to use which is detailed here.
Step 4 - Tune it up!
Using the string winder or your hands, gently bring the strings up to pitch. Don't try and do it too quickly or you can run the risk of snapping your new string without playing it - slow and steady is the key here.
If you're in a hurry, you can stretch the string a little quicker by pulling it as in the picture below - this helps take some of the stretch out of the string quickly and allows you to tune it sooner than normal, but be very gentle!
Step 5 - Enjoy!
Once your strings have settled in, plug it in and turn it up!
To keep them in optimum condition I recommend using a string lubricant like GHS Fast Fret before you play. This helps to clean the strings of sweat and other deposits and will keep them brighter for longer.