Apple - How can I activate buttons with just the keyboard?

Enable Keyboard Control of the UI

System Prefs > Keyboard > Shortcuts

Then select "All Controls" radio button at the bottom, rather than just "Text boxes and lists only".

Keyboard Shortcuts

• ⇥ (TAB) will move between buttons.
• esc (ESC) is cancel.
• space (SPACE) selects the active button (blue, outline).
• ↩ (RETURN) is OK or the default button (blue, pulsing, filled).
• For some dialog boxes, ⌘+first_letter will select the button with a certain first letter in the text (as pointed out by @Griffo).

For many dialog boxes you can select the option you want by pressing ⌘+firstletter where firstletter is the first letter of the option you want to select

To elaborate on the answers above, many (most?) dialogue boxes will have the default option in blue. Hitting ⇥ will cycle between options, with a blue outline around the other button, but the default will remain blue. Hitting ↩ will always select the default, space will select whichever button is highlighted.

In this example, pressing ↩ will 'Keep .txt' and pressing space will 'Use .bak.'