Here's how to make sure it's enabled for your users.
- Click Setup
- Do a search for "user interface" or scroll to "Customize->User Interface" (at the bottom)
- We could spend hours on each of these items, but for here let's just click "Enable Inline Editing".
- Click Save.
Now, when your users are on a record, simply double click in a field to allow changes. Those that can be changed will show a "pencil" icon to the right of the field, signifying that you can change that field without going into edit mode.
Incidentally, this works on views as well (most fields) where you can do mass updates of your records from views by double clicking the field (same pencil icon).