Different Ways to Remove a Background
Here are different ways to remove a background using the Magic Wand, Instant Alpha Eraser, Blending Modes, and Filters.
The original image in this example is asparagus with a green background. The goal is to remove the green background.

The Magic Wand Method
The Magic Wand is located under the Select Tools menu (Keyboard shortcut "M"). You will see it on the far right as shown below.

In the Select Tools menu there is a value box labeled Wand Tolerance. This number adjusts the range of colors that the Magic Wand will select. To remove all of the green background and not just sections of it, the tolerance needs to be set fairly high. A value of 60 should be enough for this image. Using the Magic Wand, select the green background. You can tell the background has been selected by all of the little ants marching around it.

Another option for fully selecting the background is to use the magic wand tool and place your cursor over the background. Then click and drag with the mouse to increase or decrease the tolerance of the selection. You will see a circle near your cursor increase or decrease in size in response to the change in tolerance. You can see in real time how your selection grows or shrinks. The first image below shows an appropriate selection of the green background. The second image shows the wand has selected too much and is overflowing into the asparagus.

Now hit the Delete key and deselect by going to Select ▸ Clear Selection or hitting ⌘D on your keyboard. This has removed the original background completely, leaving a transparent background.

The Instant Alpha Erase Tool Method
Instant alpha is located under the eraser tool as shown below.

Instant alpha is an erasing tool that has similar qualities to the magic wand selection tool. Click on the color you want erased and simultaneously click and drag outwards to increase the erasing tolerance, or inwards to decrease. The results will be seen in real time as you drag inwards and outwards.

The green background is removed, leaving transparency. This is because there is no layer below the asparagus to show through the erased green area.
The Transparentomatic Filter Method
Go to Filter ▸ Color Adjustment ▸ Transparentomatic.

Click once on the color well to the right of where it says 'remove'. This will bring up the colors palette. Click on the magnifying glass to the left of the colors display in the colors palette. Using the magnifying glass click on the green background in the image. The green background will be removed. Play around with the tolerance level while in the filter until things look right. Click 'apply' once you are done.

The Replace Color Filter Method
Go to Filter ▸ Color Adjustment ▸ Replace Color. Click once on the color well to the right of where it says 'remove'. This will bring up the colors palette. Click on the magnifying glass to the left of the colors display in the colors palette. Using the magnifying glass click on the green background in the image. The green background will be removed and replaced with whatever color you choose to have in the 'replace' color well. Play around with the tolerance level while in the filter until things look right. Click 'apply' once you are done.

The Blending Modes Method
Select your background using the Magic Wand tool. Go to Layer ▸ New Layer with Selection . The selection should transfer to this new layer. Fill in the selection with black or white, or any color, it doesn't really matter.

Now change the Blending mode of the layer. Right now it is set to Normal. To erase the background, set the Blending mode to Destination Out.

This will leave a transparent background that is easily removed or altered by deleting the layer it is on, or turning the visibility on and off.

The Color Controls Filter Method
Now use the color controls filter to remove or change the color of the background. This method will not make the background transparent. Once again, use the Magic Wand to select the background.

Go to Filter ▸ Color Adjustment ▸ Color Controls. The Color Control menu will appear. You see it only shows the background color because it has been selected with the Magic Wand, leaving out the asparagus.

These are the color controls. You can adjust the Brightness, Contrast, and Saturation of your selected background. In this case, the background color is what is unselected (the asparagus), so changing the background color will not change the color of your actual background.

To make the background white, adjust the Brightness all the way to the right, and to make the background black move the slider all the way to the left. To make the background any shade of gray, slide the saturation all the way to the left, then slide the brightness from left to right to adjust the value of gray.

With the Color Controls, by adjusting the sliders around, you can change the value and saturation of the color of your background. But you are limited to the colors of your original background. To change the color of the background to something completely different, use Filter ▸ Color Effect ▸ Color Monochrome.

In the Color Monochrome window, there are two places to select color. Again, the background color will not change your actual background color. Select a color under Color Monochrome. When you click on the color box, the color palette will appear. Choose your color! How about red? You can adjust the intensity of the color by using the Intensity slider. This will shift your color from the original background color to your new chosen background color.
Photo Credit
Courtesy of: http://www.publicdomainpictures.net