When Destiny was first released, Guardians were capped to level 20. The only way to attend some of the game's more difficult raids was to equip items that raised their "light levels" beyond level 20. Since then, things have changed in more than a few ways.
Levels don't really matter as much as light levels do. Once you hit the game's new level cap of 40, you'll still need to get your light levels up to a certain threshold before you can participate in Daily Heroic Story missions and Weekly Heroic Strikes.
You'll need high light levels to deal with some of the game's most difficult challenges. The game's new King's Fall raid has a light level requirement of 290, for example.
Light levels are basically an indicator of how powerful your Guardians's gear is. It is simply a numerical representation of your attack and defense stats across the board. The higher they are, the higher your light level. They are similar to the "item level" system in World of Warcraft.
You don't "earn" light levels, per se, as you would normal levels. Equipping different items will change your light level to reflect how powerful your gear is.
Since it's tied to gear, you'll want to equip yourself with the most powerful gear you can find. This means having a Legendary level 40 item in every slot, an Exotic weapon, and an Exotic piece of armor.
If you're just getting back into Destiny after a long hiatus, you'll want to replace all of your current items (including your Exotics) with the greens and blues you find on your way to level 40, because Year Two gear is that much better than anything from Year One. Just ignore item rarity until you start finding level 40 gear.
With The Taken King, you can now equip shells for your Ghost and artifacts, both of which raise your light level.
Always equip items that raise your light levels. The reason you'll want to do this is because of how the game's itemization system works. When an engram or an item drops in the wild, they drop based on your current light level. The better your equipment, the better gear will drop.
If you hate using blue (Rare) items, you can infuse your Legendary and Exotic gear with items with higher defense and attack values. So whatever you do, don't dismantle the blue items you find unless you're absolutely sure you can't use them to upgrade your Legendary and Exotic gear. You can do this repeatedly, provided that you find gear with higher values to consume in the infusion.
Be sure to check out our guide on the fastest way to earn Legendary Marks in Destiny: The Taken King. We also have guides on how to Infuse weapons and armor, and how to upgrade your Exotics from Year One to Year Two.