UPDATE 14/12/2015 Due to the attention this post is getting (#2 on Google search!) and the misunderstandings some people have, I want to point out what this will fix and what it won't. ALL Intel Graphics Cards cannot run any shaders pack except THIS ONE. However, very certain computers cannot run even that shaderpack. This is a tutorial on how to fix for those certain computers, normally with the driver titled Intel(R) Graphics Family. Please test the shaderpack linked above BEFORE you go through these instructions.
If you want any shaderpack to work with Intel, you are out of luck. Intel in general doesn't go too well with gaming (Minecraft is a simple game; it runs faster then others) so I suggest if you do a lot of gaming/3D rendering to upgrade your graphics card or buy a new computer.
There is a Minecraft mod called "Shaders", that allows you to make Minecraft a lot nicer. It adds realistic water, shadows, and even waving leaves! It doesn't change your gameplay, it just changed what you see. Here is a screenshot of the most popular Shader Pack, called Sonic Ether's Unbelievable Shaders:
Shader packs are all created by different people, not the Shader mod creator. It works very similarly that resource packs work.
Unfortunately, some people cannot use shaders. Those are the people with Intel graphics drivers. On your computer, you need a graphic driver to display things on the monitor. Unfortunately, the ones with Intel drivers normally don't work. The latest drivers do not support shaders, and those few computer that have an older version of Intel installed can only use some shaders made specifically for Intel. However, it is very possible to downgrade without altering your computer hardware. These instructions will only be for Windows 8 users, though it may be possible to do something similar to what you see here for older versions of Windows.
First of all, you want to open the Device Manager. You can do this by clicking the Windows key and X together. Then, you want to click Device Manager. Now, you want to click the little arrow to the left of "Display Adapters". For me, there is only one item in the list. Mine is called "Intel(R) HD Graphics Family". If you have more than one item in your list, look for something similar to that. Now, you want to right click it > Properties. Click to the tab titled "Driver". Before you continue, you will want to download Intel driver that you need for shaders. Click here for the download page. There is a download button on the page.It may take up to five minutes. When it is done, extract it into it's own folder. I recommend putting it in it's own folder in the Documents folder. Now, you want to go back to the properties window of the Intel graphics driver. Then, click the "Roll back Driver". The screen WILL GO BLACK for a few seconds, this is normal. And now you click the update driver button. Select "Browse My Computer For Software", find the folder with the driver you downloaded and select that. Now, you want to click "Next". It should start installing the driver. Again, the screen will go black for a few seconds.
And now, you have a shader-compatible Intel driver! Now you want to install the shader mod. Click here for the mod page. Once you download the mod (make sure you have forge for 1.7.10 installed), put it in your mods folder in .minecraft, then start Minecraft with Minecraft Forge. It shouldn't take too long. Then, click Options, and (next to Language) click Shaders. Then, click "Open shaderpacks Folder". It will open the folder in another Window. Then download the shader pack here. Now, you want to extract it and copy the Ultra folder into the shaderpacks folder. Go back to Minecraft, and select the shaderpack from the list.
And that's it, you now have shaders!