Champions generally deal damage either by auto-attacking or by using abilities. Depending on the champion you are playing, you should by only Attack Damage items or Ability Power items. To find out if the champ you chose is AD or AP, look at the info page.
Each champion has abilities which usually scale off of AD or AP. Sometimes the ability scales with other stats like Health, Armor, Magic Resist, Movement Speed, etc. For new players, I recommend a champion with targeted (point and click) abilities and self/ally buffing abilities. Skillshots (aim and shoot abilities) take practice and you should focus instead on understanding the game's items and champions.
If you're looking for champions that can be quickly learned I'll list a few.
AP Champions:
-Morgana
A little trickier, as her main source of damage is a skillshot, her Q that also roots what it hits. Her W is an aoe pool that deals damage to those who step on it. E is a shield that soaks up magic damage and stops ALL disables (slows, stuns, knockups, etc.) that affect you while it's up. Her ult deals damage to nearby champions and 3 seconds later deals that much damage again to those still near you and roots them.
-Annie
Easier to pick up. Q targets, W is a short ranged aoe skill shot, E is a self buff that reduces incoming damage. R is a giant teddy bear that deals a LOT of damage to anyone near where it lands. He stays around and attacks enemies for you. Annie's passive makes your next ability (after 4 casts) stun whoever it hits.
For both, build Morellonomicon, Luden's Echo, then Rabadon's Deathcap. Pick up Sorcerer's Shoes at some point. Try to pick mid lane.
AD Champions:
-Garen
Easy and fun. His Q gives you more movement speed for a short time and makes your next auto attack stronger and silence (they can't cast spells) for 1.5 seconds. His W is a self buff that reduces incoming damage. Also gives you more Health for each minion you kill. His E makes you spin you sword and deal a lot of damage. It also temporarily lowers the armor of those it hits. R is a point and click execute. It deals a lot of damage to whoever you select. It deals more damage the less health they have. Wait until they have less than 25% hp. Passive regenerates health over time.
Build Black Cleaver and then tanky items like Spirit Visage, Dead Man's Plate, and Randuin's Omen. Go top lane if possible.
-Ashe
Q makes your auto attacks deal more damage when active. requires you to attack a few times first. W is an skill shot that damages those it hits and applies your passive. E fires a "scout" to give vision along it's path and where it lands (anywhere on the map). R is a skillshot that deals damage to the first champion hit and stuns them depending on how long it traveling. Slows nearby enemies upon impact. Passive applies a debuff to enemies that you hit with auto attacks and also your W. The debuff makes your auto attacks deal more damage (scaling on your critical hit chance) Also, you can't crit for more damage, you just always deal more damage instead.
Build Runaan's Hurricane, Infinity Edge, Static Shiv, and Berserker's Greaves. Try to play in bot lane.
All these champions are cheap to buy as they each cost 450 ip (except morgana, she costs 1350).
As for runes/masteries, don't worry about them. They are being replaced soon. Worry about the new stuff after you get to level 20 or so.
Regarding the jungle; that's the area in between lanes. There are monster that give gold in there if you kill them. Generally there is 1 person top, 1 mid, a support and a ranged AD champion bot, and someone in the jungle. The "jungler" gets gold by killing the neutral monsters in the jg. They also "gank" the other lanes when they can. "Ganking" is going to another lane to try and kill a low health enemy champion. Usually the jungler will only gank other lanes if it's safe (the enemy is close to your tower).
Also, FYI: ~20 cs (minion kills) is equivalent to one champion kill. So, the person who is 3/1/0 with 20 cs has a LOT less gold than the person who is 1/1/0 with 80 cs. They have about 600 less gold.
Have fun learning the champions and have even more fun playing!
[slayer-jinx-wink]