The influx of new players that came with the official release of Path of Exile has given way to a ton of questions, mostly about Path of Exile items like gems. Understanding the value of gems in the game is a basic knowledge that opens up to knowing more about the rest of the game as they are crucial to every player. It does get quite confusing for beginners though, so here are some fundamentals regarding non-quality gems and their value in the Standard league. After reading this, whenever you get to choose which one to get after a quest, you'll know which one to take.
Higher Value Gems
Most gems are gained through quests, but some only come from monster drops. Since they're rarer, they demand higher prices. Portal gems are drop-only, but they're not that useful since Portal Scrolls are not scarce to begin with. They can be sold for 1-3 Chaos Orbs though. Chain gems are only useful for a few builds, so they cost just 1 Chaos Orb.
Cast When Damage Taken gems are a reward from a single quest in the game, which is "A Fixture of Fate" in Act III Merciless. They can also be obtained from drops, but they are so rare that it's only feasible to have one per character. They do get traded a lot, and having a level 31 requirement means that you can't just make use of it right away. They are usually priced at around 5 Chaos Orbs.
Multistrike gems are similar to Cast When Damage Taken gems, also coming from either "A Fixture of Fate" or monster drops. But since they're used for just about every melee build in the game, they are in demand. But due to that demand, players look for them and the supply has grown in abundance. In the end, they cost only 3 Chaos Orbs each.
Other high value gems include the Cast When Stunned gem, which is at around 2 Chaos Orbs but can be easily obtained by yourself; Empower at a high 10-20 Chaos Orbs; and Enhance at 8 Chaos Orbs. The latter are both rare and in-demand, which is why they're so expensive.
As for the other gems, you still need to know what they are to not confuse them with the higher value gems and to see if you can make use of them as well. Just because they're not high value, it doesn't mean that they're trash PoE items. For instance, Auras are always sought after and can be sold for 1 Orb of Fusing, or even 1 Chaos Orb if you're lucky. On the other hand, Grace, Hatred and Clarity are some of the easiest ones to sell in the game.
Most attack and other offensive gems are very common, so you may find it hard to sell those since just about everyone has them. Curses can be sold for 1 Alchemy, but only if someone is looking for a specific one. If you're alright with keeping a few dozen of them just in case, then you're free to do so unless you have to use that time and storage space for something else.
Support gems are the ones that get traded the most, with most of them going for 1 Chaos Orb each. Then there are the support gems that don't get as much utility, which can go for 1 Fusing for each one. That about covers the basics with the gems, and everything else is just stuff that you'll encounter as you play the game more.