Alternative costs allows players to play the card by different means, than playing than mana cost. This often includes certain advantages, mostly these cards have a lower alternative cost compared to their original mana cost. Some creatures with suspend also have additional effects by being suspended (e.g. Aeon Chronicler). Morph creatures, also, often have effects when morphed. Note however, that not all Morph costs are lower. E.g. Brine Elemental's Morph cost is one higher, than his mana cost. However, Morphing him has the additional effect of tapping the opponent's creatures.
Some cards (e.g. Fireblast) have the alternative cost written on the card. These types of alternative costs will be explicitly mentioned on the card.