What's the difference between Presta and Schrader, and will these pumps fit both?

Presta is the thin valve common on road and higher-end bikes; Schrader is the wider automotive-style valve. Every pump in this roundup is advertised as dual-compatible, and reviewers single out smart or twin-valve heads (like BV's reversible head and Topeak's TwinHead) for switching without leaks or fiddling.