There are two places that field permissions can be set, and these two settings are mutually exclusive (changing one does not affect the other).
 
App Setup
  1. Go to Setup -> App Setup -> Customize
  2. Choose the object type (Accounts, Contacts, Opportunities, etc) and click Fields under that object
  3. Find the field that needs permissions set and click the name
  4. Click "Set Field-Level Security"
  5. Find the profile or profiles that need permissions added and check or uncheck Visible and Read Only
Visible will allow the field to be available and Read Only will limit to being only Visible.
 
Layout
  1. Go to any record of the record type needing to be changed
  2. Click Edit Layout in the top right corner
  3. Find the field that needs to have permissions granted Note: If the field is not listed in the layout, then it must be found in the palette above
  4. Hover over the field and click the wrench icon
  5. Check or uncheck "Read Only"