The aim of this thread is to provide a bit more fleshed out information for players about glyphs, and the reasons that people pick them. It's not going to be as detailed as some of the advanced theory crafting threads found elsewhere, but will explain a bit more about how they work in game.
First off, just like talismans and weapons, glyphs have what are commonly referred to as pips. 
Glyphs have crude (one pip), simple (two pips) and intricate (3 pip) variants. The amount of any stat a glyph gives is related to their pips. Intricate gives 100% of a stat, Simple gives 90% of the stat and Crude gives 80%. That means that to get the highest possible stats, you'll need all of your glyphs to be Intricate.
This is only really important once you get to level 50, and are starting to fuse glyphs into the higher rarities (green<blue<purple<orange<red). Unlike talismans and weapons, glyphs always max out at level 20, and you need to fuse identical max level glyphs to bump them up to the next rarity.
Location, location, location?
When glyphing talismans, it doesn't make any difference which glyph you put in which talisman. A head talisman glyph will give the same stat increase as if it were placed in a waist talisman for example.
Weapon glyphs are slightly different. You only gain the effect of the glyph in the weapon you are using for an ability. So if you have a lvl 20 glyph in your primary weapon and a lvl 1 glyph in your secondary weapon, you will only have the stat boost of your lvl 1 glyph when you attack or heal with your secondary weapon. The most common approach is to use the same glyph type in both weapons, as that way your stats remain even when fighting. It's still better to level up your primary weapon's glyph first though, as it will be providing you with the majority of your damage. Because of this, many players choose to level their talisman glyphs first, then do their weapons, as that way you are boosting all of your damage, not just a percentage of it.
Glyph types:
Fierce- provides critical rating. Critical hits deal more damage or healing, the additional amount will vary depending on your critical power rating.
Devastating- provides critical power rating. Critical power rating increases the additional damage or healing delivered by critical hits.
Accurate- provides hit rating. Reduces the chance your attack will glance on a target. Glancing means you deal massively reduced damage, and any interrupts which your ability might trigger will not work.
Elusive- provides evade rating. Evading attacks totally avoids all damage and effect which the attack might trigger.
Stalwart- provides defence rating. Defence rating increases the chance that your enemies attacks will glance on you.
For tanks and DPS, the first thing to prioritise is hit rating. Glancing drastically reduces your damage, and stops you interrupting bosses, so it's important to get your hit rating up. As you level, you may notice that your attacks glance more often. On your character sheet, you'll see a stat called "glance reduction", and this is what hit rating will improve. You don't want too many Accurate glyphs though, as a single legendary (red) Accurate glyph is all you will need for elite 10 dungeons. What works for many people is to use 2 Accurate glyphs, which then can be fused together to make the one legendary later on in the game. Only one needs to be intricate, the second can be solid and you'll still have a good chunk of glance reduction.
Once you've dealt with glances, the next priority is Critical rating. Fierce glyphs are the most frequently used glyph, which means they're also the most expensive. It's the main way we're able to boost our damage, and big numbers are always nice to see. Critical hits & heals also generate one energy for the weapon which crit, which means that with high crit rating you will find yourself much less energy starved. It's a fairly long term investment though - you won't really be swimming in crits until pretty late in the game. In fact, even if you ignored hit rating and just dumped everything into crit, you're still not going to get fantasticly frequent crits.
It's normal for players fresh to elites to go with 5 or 6 Fierce glyphs on their gear. If you're using 2 Accurates, you'll only have room for 5 Fierce, but you can also choose to take just 1 Accurate and take a 6th Fierce - you just need to work hard to keep your Accurate glyph as levelled as possible.
Next up is Devastating glyphs and crit power. You start off with a basic +100% critical damage, (so crits will deal double damage or heals) and Devastating glyphs will improve that. It's a fairly low priority stat though - after all, if you glance it doesn't matter how much crit power you have, and you need to have got a crit in the first place to get any benefit from your Devastating glyphs.
The good news is that because Devastating glyphs aren't as extensively used as Fierce, they're normally cheaper! Because of this, it's a common choice to put Devastating glyphs into weapons. If you're going to put the same glyph into all your weapons, that means that you'll want between 2 and 9 Devastating glyphs, (depending on how many weapons you have). That works out a lot cheaper than having to buy extra Fierce glyphs! You'll also see less variation in your crit related energy generation for different weapons, which makes a rotation easier to practice.
Ultimately, you'll want 3 Devastating glyphs - 2 on weapons and 1 on a talisman. There's maths involved about why this is, if you'd like to know more, let us know and we can point out the relevant threads elsewhere.
Er, what about healing?
The good news is that healers only need to swap out any talismans with Accurate glyphs to be fully focused on healing. If you're using an Assault Rifle to heal, then you'll still want glance reduction though, so AR uses the same glyphs as DPS. You can heal without changing anything, but if you have a dedicated heal set, you can just replace any Accurate glyphs with Devastating ones for bigger critical heals.
So tanks use the defensive stats, right?
Um, not really. The current state of tanking is that you can use Elusive and Stalwart glyphs, or you can just use the same set up as DPS. If you decide that you want to go with defensive stuff, then it's better to pick just one type. They do help keep tanks alive longer, but not very much. It's kind of like having a motorcycle crashing into you whilst walking, instead of being hit by a car - there's a chance you'll survive, but unless you've got something else protecting you, you're still going to be pretty badly hurt. Defensive cooldowns are necessary to avoid being flattened by bosses regardless of your glyphs. That means you have the choice to use defensive or offensive glyphs. With the crazy numbers that well geared DPS can achieve, many (most) tanks prefer using crit rating and damage too, to help them generate more hate and hopefully, hold aggro. It also makes it easier to level - you'll kill stuff faster when levelling with offensive glyphs, and you'll be able to switch between tank and DPS roles without penalty.
What about while levelling, isn't a bit of defence useful there?
The short answer is no. It's better to take an all out approach, where the best defence is to kill your target before they kill you. All the weapons except Elementalism have some kind of self heal available. You'll find it faster to go full damage and use one (or more) of those, rather than mixing in a few defensive glyphs. You only really get the full benefit of defensive glyphs when you have a lot of them, just one won't help too much.
So the normal glyphs to use in low elite levels are:
1/2 Accurate glyphs
6/5 Fierce glyphs
2 Devastating glyphs (on weapons)
Once you're into higher elite levels:
1 Accurate
5 Fierce
1 Devastating glyph on a talisman and 2 on weapons (so 3 glyphs total)