Switch to Material viewport shading by clicking the icon in the top right of the 3D view:
![Textured Suzanne with material shading enabled](https://i.sstatic.net/6hLNf.png)
If you have an image texture, you can also display textures in solid shading mode by switching to "texture" in the shading mode options drop down:
![enter image description here](https://i.sstatic.net/DrD2o.png)
Blender 2.5x - 2.7x:
Blender internal:
Try setting the viewport shading to Texture (AltZ):
![enter image description here](https://i.sstatic.net/bPEph.png)
Note that this only works when the object's UV map has the appropriate texture displayed in the UV/image editor: (see this post for more information)
![enter image description here](https://i.sstatic.net/lBwi5.png)
To display the textures used in the material of the object regardless of the texture (or lack of texture) displayed in the UV image editor, enable GLSL in 3D view > Properties region (N) > Shading and set the viewport shading to Texture:
![enter image description here](https://i.sstatic.net/R1nmf.png)
Cycles:
You need a material to display textures in the viewport in cycles.
Set the viewport shading to Texture or Material.
Texture displays the texture of the last selected image texture node:
![enter image description here](https://i.sstatic.net/6FcQe.png)
Material tries to approximate the result of the shader nodes, so if you have two combined textures:
![enter image description here](https://i.sstatic.net/Uk8Ut.png)