Select the object or image you want to crop
Place a shape over the area you want to keep
Select both the shape and the image
Go to Object
Click Clipping Mask
Choose Make
Use the Direct Selection Tool to adjust the crop if needed
For raster images, use Crop Image from the top control bar
Adjust the crop handles
Press Enter or click Apply
For vector artwork, use the Pathfinder panel
Select the shapes you want to crop
Click Divide, Trim, or Minus Front as needed
Ungroup the result if necessary
Save your file