Safari

home page, Apple's documentation for using the Web Inspector Panel

keybinding: ⌘⌥i

Select Preferences in the Safari menu

Check the Show develop in the menu bar option

Click Show Web Inspector in the Develop Menu

Google Chrome

home page, using chrome developer tools documentation

keybinding: ⌘⌥i

Click Developer Tools in the wrench menu, tools, and Developer Tools

Mozilla Firefox

home page

keybinding: ⌘⌥i

Download Firebug at http://getfirebug.com/

Start Firebug at Tools / Firebug / Open Firebug

iOS 5.0 Mobile Safari (iPhones & iPads)

for advanced debugging, see weinre

Touch Settings on the home screen

Touch Safari on the settings screen

Touch Advanced on the safari screen

Toggle debug console to true

A debug bar will now appear at the top of each webpage in mobile Safari.

iOS 6.0 Mobile Safari (iPhones & iPads)>

Touch Settings on the home screen

Touch Safari on the settings screen

Touch Advanced on the safari screen

Toggle debug console to true

Connect your Device to a Computer

Select Preferences in the Safari menu

Check the Show develop in the menu bar option

In Safari on a Computer, select Develop, and then the appropriate Device

Internet Explorer

Discovering Internet Explorer Developer Tools

keybinding: F12

One option is installing the Internet Explorer Developer Toolbar from Microsoft. Firebug Lite is a decent, open-source option. DebugBar is a closed-source option that's slightly more complete than Firebug Lite.

Opera

home page

keybinding: ⌘⌥i

Tools → Advanced → Opera Dragonfly