Toughness is a characteristic of creatures, it is the number on the right side of the box in the bottom right corner of a creature card. Toughness is a measure for how much damage a creature can take before it is destroyed. A creature is dealt damage through combat, spells, and abilities. Creatures battling will each deal damage equal to their Powers to the other creatures toughness. If a creature has equal or more damage marked on it than its toughness, then it is destroyed.Many spells can modify a creatures Toughness or Power.
A toughness of * is defined by abilities of the card.
A 4 power and 4 toughness (4/4) Boonweaver Giant attacks, and a 2 power and 5 toughness (2/5) Totem-Guide Hartebeest blocks it. After combat damage is dealt the Giant is a 4 toughness creature with 2 damage marked on it and the Hartebeest is 5 toughness creature with 4 damage marked on it. Since neither received damage equal to or higher than their toughness, they both survive. (You can, however, play another spell to finish off a creature that has already taken damage after combat.)
A 3/1 Deadbridge Shaman attacks, and a 2/2 Tower Geist blocks. After combat damage is dealt, the Shaman is a 1 toughness creature with 2 damage marked on it and the Geist is 2 toughness creature with 3 damage marked on it. Since they received damage equal to or higher than their toughness, they both die.