When asking how to better survive, the first question is, what exactly is out there trying to kill you?
Problem 1: Harass
A lot of champions have really annoying harassment that slowly but steadily reduces you to below half health, at which point they come in for a kill. Sometimes this comes in the form of skill shots (Lux's Lucent Singularity, Ezreal's Mystic Shot), other times it can be auto attacks (like most AD carries), or even some point and click abilities (Akali's Mark of the Assassin, Jax's Leap Strike). The common factor is one way or another, they keep wearing you down to the point where they can easily kill you.
Solution 1: Sustain
There's three different things you can do against harass-heavy lanes. The first is to buy some sort of armor/magic resistance so their harass does less damage. Against heavy physical damage you might take cloth armor and 5 potions as your starting items, for instance.
The second thing is buying some sustain. This means life steal (for AD champions), spell vamp (for AP champions), or health potions (for everyone). These items give you back some health over time, and if you're farming well can make staying in lane much easier. If you find yourself buying a lot of health potions, consider grabbing a Crystalline flask, as it'll be useful all through the early-mid game, and is definitely more gold efficient if you remember to use it.
The third and final thing is to simply recall sooner. Yeah, it sucks being forced out of lane, but it's better than giving your lane opponent another kill. Sometimes you're best off admitting defeat and recalling, so you can come back with a full health bar, a couple new items, and a chance to turn the tables on your opponent.
Problem 2: Burst
So you're just walking along in bottom lane at high health, last hitting creeps, when all of a sudden the enemy Blitzcrank turns and rocket grabs you. Meanwhile their AD carry continues shooting you with auto attacks and a couple of abilities, and before you can run away you've died yet again.
Or perhaps you're up in top lane against Riven, and she activates her ultimate, dashes to you, stuns you, uses her broken wings/auto attack combo, before finishing you off with an ignite and wind slash. From full health to death in just a few seconds, and you have no idea how to stop it.
Solution 2: Defense
If the enemy laner has enough power to take down 80% of your life (or worse, all of it) in a single combo, sometimes all you can do is put your normal build on hold for a mid level defensive item so you can survive. If you're up against AD damage, you'll likely want a Chain Vest, or possibly a Warden's Mail. If you're going against magic damage, a Negatron cloak is a good buy. (AP mids might also build Chalice of Harmony into Athene's Unholy Grail.) A Giant's belt is a good catch-all buy for the extra 380 health, and can be later finished into a Warmog's armor (or some other useful item) for even more survivability.
If crowd control is a problem, you can choose Mercury's treads as your T2 boots (they're generally amazing anyway), and they even reduce magic damage as a bonus. Conversely, Ninja Tabi is great against physical damage, particularly auto attacks.
Finally, if the enemy's combo depends on them landing a skill shot (Blitzcrank's Rocket Grab, Lux's Light Binding), then get your tier two boots as soon as possible, and stay behind your minions (if they block said skill shot). The fact of the matter is, there's a lot of instances in League of Legends where getting hit by a skill shot will lead to your death, so the only thing you can do is practice dodging them. Whatever you do, don't stay still, and don't run in a straight line.
Problem 3: The Jungler
Last (and possibly most annoying), we have the Jungler. This is the guy that jumps out of the bushes on the side of your lane to murder you, stopping your onslaught against your lane opponent, and ruining an otherwise decent laning phase. (Of course, other enemy champions aside from the jungler can do this, he's just the most likely to ruin your day.)
Solution 3: Sight Wards
The fact of the matter is, most champions in league of legends are not equipped to fight two opponents at once, and trying to do so is a recipe for disaster. The best thing to do is avoid fighting the jungler, and the best way to do that is by seeing them coming. A mere 75 gold investment (or 150 gold if you're the mid laner) is enough to ward the bush(es) by the river, and keep their jungler from surprising you. If you see him in the bushes, fall back and wait by your tower. Those few minion kills are not worth your life.
If you don't have any wards left, and you know their jungler is out there, play back in the lane. A good rule of thumb is that being on your side of the river is fairly safe, being about even with the river is risky, and being on the enemy's side of the river is very dangerous.
Finally, keep in mind that a couple of champions have ways to bypass your wards entirely. Someone like Twisted Fate or Pantheon can just teleport on top of you, and turn what seemed like an advantage into your death. Be very careful about engaging an enemy champion if one of those guys has their ultimate available.
Other things you can do
Remember, League of Legends is a team game, and you have allies is well. Sometimes the enemy champion counters yours, or they got an early advantage, or they're just outright better than you. If that's the case, ask for help! A couple of ganks from your jungler (or even other laners) can force them to play back a little further and respect your space a little. In emergency situations, you can have someone else take over your lane, and try your luck somewhere else. Remember, it doesn't help your team if the enemy laner 5-0's you in the first 12 minutes, and if they can do anything to help out, they definitely should.
Also, keep in mind that the laning phase is not the end of the game. Some champions focus on the early game, dominating their opponent and setting the tone for the rest of the match. However, other champions only become truly powerful in the late game, and tend to have difficult or risky early laning phases. (See: Veigar). Don't give up just because the enemy team is fed, as good teamfighting can turn around a lot of bad situations, and that 5-0-3 Pantheon might just end the game 6-8-7.
Last but not least, sometimes the enemy team is simply better than you, and runs over you and your allies at every turn. Don't get discouraged, it happens. There's an incredible number of champions in League of Legends, most of which are quite devestating under the right circumstances, especially if you don't know how to handle them. If you're having a lot of trouble with a certain champion, check a guide on them (to figure out their abilities), try a few different champions against them, and ask around for help. It might still be a struggle, but there's a way to deal with everything in this game, if only you can find it.