I don't know what driver to update or how? Graphics rendered glitchy in some games?

I'm having precicely the same issue shown in this youtube video:

The textures (primarily for the foliage, but also for just about everything else in certain places) is stretched obnoxiously over all of Skyrim, obscuring my view of paths and, more problematically, dragons. This issue doesn't seem to occur much in individually loaded areas such as caves or buildings, but the more vast the space the more render issues there are.

This also occurs with Sims 3 and Goat Simulator - but not Minecraft, Portal, Portal 2, Amnesia (either of them), or Bioshock.

I've tried googling the problem and most of the solutions involve "Updating to the most current drivers" - which may sound straightforward enough for people who know how to do that, but I'm really utterly clueless about this sort of thing. Which drivers should I update? Where do I go to update them? How do I do it?

I'm not sure what information really makes a difference here and what doesn't, but here's what I know: I'm using Windows 7, not 8, and I'm using an HP Pavilion dv7 laptop.

Added (1). Actually my issue isn't quite as it is in that video. Any one still from that video matches my issue, but the odd renders don't change anywhere near that rapidly.
Additionally I'd like to know if outdated Java could have some influence. I do not want to update my Java unless absolutely necessary ^^;

So, how does your Video Graphics Card with its dedicated memory stack up when you check the various game's requirements?
EXAMPLE
Skyrim Specs: (Video Graphics is where most machines fail the test)
You Do need a machine that meets these requirements for Skyrim.
Recommended
-Windows 7/Vista/XP PC (32 or 64 bit)
-Quad-core Intel or AMD CPU processor
-4GB System RAM
-6GB free HDD (Hard disk drive) space
-DirectX 9.0c compatible NVIDIA or AMD ATI video card with 1GB of RAM: Nvidia GeForce GTX 260 or higher; ATI Radeon HD 4890 or higher
-DirectX compatible sound card
-Internet access for Steam activation