Why does Minecraft grass have different colors, and how can I make it bright colored?

Grass is different colours in different biomes, and you're seeing the "dry desert" grass colour in the middle of your screenshot. The difference in colours is controlled by one texture file though, so it's easy enough to fix by installing a resource/texture pack that changes that file to be one uniform green instead of a gradient of greens and yellows.

Texture packs like that used to be popular due to nostalgia for the grass in older versions of Minecraft, but my googling just now reveals that they're a harder to find now. Since googling turned up nothing except broken links, I threw one together:

Green Grass Texture Pack

If you want a different green than the one I picked and have some facility with zip-file creation and image editing software, you can find the one texture file it contains and change the triangle of green to a green more to your liking.

Installing a Texture Pack

1. Open Minecraft
2. Click Options… then Resource Packs…
3. Click Open resource pack folder
4. Switch away from Minecraft to your desktop (if it didn't automatically)
5. Copy the downloaded file in to the resourcepacks folder that Minecraft just opened
6. Switch back to Minecraft
7. The new resource pack won't show up until you close the Resource Packs screen and reopen it: click Done then click Resource Packs… to open it again
8. Select the new resource pack and click Done

Grass color is based on biome. The grass in the middle is in a desert biome. You can't really control this without just picking another spot (I guess if you really wanted to you could change the biome with MCEdit).

Jungle biomes have the brightest green grass, forest and plains biomes also have fairly green grass. Pretty much any biome except the particularly arid ones will have grass like the grass surrounding the brown grass in your screenshot (with the exception of swamp biomes, which features really dark green grass).