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enemies are 4 or more levels higher than you, which means that you cannot damage them, only anger them.
Eventually you'll get high enough level to see their actual level and be able to damage them.
The soft level capacity is at 20, yes. That means when you reach level 20, experience will no longer make you level any more. If you want to go past level 20, you'll need Motes of Light to buy equipment from the Speaker. This armor will have a stat called Light which determines how many more levels past 20 you can get to. Max level 32. This increases when new content is added to Destiny.
EDIT:
The class items bought from the Speaker does not give any stats. So the only way to get equipment with the Light stat is to kill enemies and hope for good equipment with the light stat. Only higher level equipment will have the light stat.
EDIT 2 (Feb 6, 2015)
Motes of Light are given to you when you accumulate experience after you've reached level 20. Equipment with the Light stat can be bought from the Vanguards and from Xûr. It can also be awarded after missions, strikes, raids, and battles in the Crucible/Iron Banner. The rarer the equipment, the more light.
UPDATE: (May 19, 2015)
With house of Wolves out now the new level cap is 34. You can use etheric light to ascend (upgrade) any of your old legendary or exotics though to catch up with the new gear and level cap. Get more info here: http://goo.gl/VPztvk
Update: (September 9, 2015)
With Update 2.0.0 the Character Level Cap will be increased to 40 on 9/15/15, and you no longer need armors' light stat to further increase your effective level. It's now purely EXP based. There is also now a gear score component, where each equippable gear piece's defense or a attack level contributes to your overall light level. This maxes out at 310, and is based on the average value (possibly with some weighting) of all of your currently equipped gear.
Update: (April 12, 2016)
Bungie released their Update 2.2.0 on April 12th, 2016. The Light Level was increased to 335; the maximum XP level still caps at 40
Update: (September 20th, 2016)
In the Rise of Iron Expansion, Bungie increased the maximum light to 385
Best Answer
With the release of The Taken King, light level and character level have been separated and the progression system has been redesigned.
Character level now progresses all the way to 40, rather than stopping at 20. Continuing to level beyond 20 is simple - you gain experience points as normal. The main point of character level is to enable you to equip more powerful gear, which directly influences your light level.
Light is now the average of all your attack/defense values (in the 2.0 update, Attack and Defense values were rebalanced to reflect this). Your light level progresses as you acquire better gear, and is now affected by your weapons as well.
So, how do you obtain stronger gear in The Taken King? There are a few ways: