Gems

Like most 'free' games which involve leveling up there is a premium in game currency which can be purchased with real money, and also earned (slowly) within the game.

Progress can be slow without gems and the game tries to encourage you to buy gems, as it is a free game and the only way the developers will make money from the game. Personally I think supercell should be more generous with gem distribution. 
Progress can be slow without gems, especially by the time that upgrades take days to complete, you have already invested a lot of time in the game when you realise you need to spend money or progress  slowly. You start the game with 500 gems and you should try to save as many as possible for later in the game.
Builders Huts are the most useful item that can only be purchased with Gems and buying an additional builders hut is a smart way to spend gems if you want to progress quickly. (The only time I have paid for gems is to buy an additional builders hut).
Spend your gems sparingly especially if you're in the early stages of the game as you will need them more later, if you can wait for something to finish upgrading or try to raid instead of buying more gold/elixir. If you don't want to buy gems then patience is key, the game requires so much time an upgrading that you just want to upgrade constantly and as quickly as possible but it is much better wait and enjoy the game or before you know it you can spend a lot of money on the game.

To cut just one day off building time will cost you 260 gems, and buying resources is generally not worth it in most cases. Check out the gold and elixir section and read about farming as a quick resource solution.

GardeningClean up your village and get rid of trees and bushes can earn you gems. You get between 0 - 6 gems each time.

AchievementsThe main ways to gain gems are from completing achievements which you can see by clicking on the icon of this lady:

The most lucrative achievement which everyone should aim for is this one:

Sweet Victory!

Achieve a total of 75 trophies in Multiplayer battles: 10  / 5 

Achieve a total of 750 trophies in Multiplayer battles: 100  / 10 

Achieve a total of 1,250 trophies in Multiplayer battles: 1,000  / 450 

it is worth 450 gems and gaining trophies isn't too hard when you get good attack and defence systems going. The method I used for this was a good defence, plant bombs where the attack is coming from and re-arm them, make sure your clan castle is always full as matchmaking is based on your trophy count. If you're at a lower level than most people with your trophy count it won't take long for you to be attacked, let them attack you while you have as many bombs and traps planted as possible and a full clan castle, then as soon as you have been attacked request troops and re-arm bombs and traps, this worked for me and I got the achievement at quite a low level. Look through the achievements and see which ones are achievable for you.

The new clan wars achievements offer some good rewards so join a good, active and generous clan.

War Hero

Score 10 stars for your clan in War Battles: 50  / 50 

Score 150 stars for your clan in War Battles: 500  / 200 

Score 1,000 stars for your clan in War Battles: 1,000  / 1,000 

Spoils of War

Collect 800,000 Gold in Clan War bonuses: 100  / 25 

Collect 15,000,000 Gold in Clan War bonuses: 500  / 100 

Collect 80,000,000 Gold in Clan War bonuses: 5,000  / 1,000 

There are 66 Achievements worth a total of 9,432 Gems! so go Achieve!

Hacks/Cheats

It's also worth noting that there is no hack available for free gems, and probably won't ever be. we've all searched for them but I doubt one will ever exist, especially as the game is one of the top grossing apps in the app store.

The jailbreak tweak; iAppCracker which sometimes unlocks in game addons doesn't work for this game either. If one did appear (one did for a short period of time for simpsons tapped out) it would quickly be stamped out.

Any site asking you to do a survey for access to free gems is automatically fake!