OK, on this one, we're assuming you have a Tun-O-Matic bridge. This is what's generally installed across a huge swath of the Gibson range and it's almost certainly what's installed on your Les Paul, SG, Firebird, Explorer, 335, etc., etc. There are a couple of styles of tun-o-matic but the basic operation is the same. If you've got a wraparound-style tailpiece bridge, check out that article.

The adjustment screws can be a bit fiddly to access so be careful. Try not to damage your strings, your guitar top or the screw itself. Use an appropriately sized screwdriver (usually a small-medium flat-blade)

HOW TO SET INTONATION ON A GIBSON TUN-O-MATIC

First up, though, let’s recap the prerequisites.

The rest of your setup must be right for you before you start. Intonation is the last thing to set so get your action, relief, nut and pickups sorted out first. You should have fresh strings (of your usual gauge and brand) installed, properly stretched, and tuned up as normal.

Remember, too, always check intonation and tuning with the guitar in the playing position (i.e. not lying on a table or counter but upright as if you were playing it).

The basic theory is this:

Pick the open string and verify it’s in tune.
Fret at the 12th fret and pick this note. Compare it to the open string—is it flat or sharp?
If the 12th fret note is flat, move the saddle forward a little by turning the adjustment screw. You'll usually want a small, flat-bladed screwdriver for this.
If the 12th fret note is sharp, move the saddle back a little by turning the screw.
Retune the open string and go back to 1.

Setting intonation on a Gibson tun-o-matic

GIBSON TUN-O-MATIC INTONATION QUIRKS

Usually, the adjustment screws are accessed from the back of the bridge. Occasionally, though, you might find a guitar with the bridge reversed—the adjustment screws are accessed from the front. This can be a bit tricky—drop your bridge pickup down a little if it helps get to the screws and be careful not to scratch or dent the pickup.
Because the tun-o-matic is pretty narrow, and there’s not a lot of back/forward movement possible, Gibsons may be less forgiving to certain string gauges. Lighter gauge strings can require more overall compensation. If you’ve problems intonating your 8-gauge strings, you might have to flip saddles or move bridges (see below).
You can see that the Gibson bridge is installed at an angle—farther back on the bass side—in a similar manner to an acoustic guitar. Since the tun-o-matic bridges are generally relatively narrow, this gives them a ‘head start’ in the compensation game but sometimes it's not enough. The saddle flipping trick outlined below might help.
Every now and then, you might come across a Gibson that just will not intonate. Sometimes, the bridge is just in the wrong place (seriously—it happens) and, even if you flip saddles, there isn't enough travel to intonate. If you bought it new you can try warranty service. There are also some after-market tun-o-matic bridges that are wider and might allow extra travel if you can find one. And, as a last resort, you can have the original holes plugged and the bridge re-mounted. It's a pain but it's occasionally required.

REVERSED SADDLES ON A TUN-O-MATIC

The big quirk on these bridges is the reversed saddles. It might be helpful if I expand a little on the bridges themselves.

While there are any number of slight variations, there are two types of bridge fitted to most Gibson electrics. They are the ABR-1 and the Nashville.

The ABR is the original bridge. Usually, you can look at an ABR bridge and see that two or three bass-side saddles are installed ‘backwards’ (with the sloped or angled side pointing towards the neck). This is to make up for the fact that there is insufficient travel on the intonation adjustment screws to get these strings to properly compensate. By flipping the saddles, you can get a tiny bit more compensation before the saddle hits the back of the bridge. Essentially, it’s a cludge. It’s a workaround.

Gibson tried to address the issue by introducing the Nashville tun-o-matic bridge. This bridge has a little more travel and it generally intonates without any saddle-flipping. Of course, as with many changes made by Gibson, lots of players complained they preferred the original and so you’ll find a mixture of Nashville and ABR bridges installed across Gibson models.

FLIPPING SADDLES ON AN ABR-1 BRIDGE

The ABR already has some saddles flipped. The Nashville doesn’t. However, on either bridge, you’ll occasionally have to flip one yourself. Sometimes you’ll be at the end of the saddle’s travel—it won’t go any farther back or forward but you’re not quite intonated. Flipping a saddle might just get you where you need to be.

On the ABR, it’s pretty easy (if a little fiddly) to do.

First up, slacken off the string you’re intonating.
On the bridge, there’s a really thin ‘retainer wire’ holding the saddle screws in place and stopping them from falling out. Use a small flat-bladed screwdriver to remove one end from its hole and carefully move it so the screws are free.
Lift the saddle and screw out of the bridge slot. You might need to prise it slighly. Be gentle.
Back out the adjustment screw, flip the saddle 180º and re-insert the screw.
Pop it back into the bridge and (the fiddly bit) put the retainer wire back in position over the saddle screws and get the end back in the hole.
Bring the string back up to pitch and continue intonation.

Reversing a saddle on a Gibson ABR-1 tun-o-matic bridge

FLIPPING SADDLES ON A NASHVILLE BRIDGE

On the Nashville bridge, you can get lucky and find a model that the screws just ‘back out of’ but most often the screws are ‘captured’ by a clip of some kind.

This may be a circlip (C-clip) or wire clip. Getting these off to flip saddles is possible but it’s a lot of hassle—especially putting them back again. The wire clip in particular will probably need to be replaced afterwards.

Approach this with caution and talk to your trusted repair person rather than forcing the matter.