A cycle is a series of cards that is specifically designed to fit together in a way. They usually have the same Mana Cost, if they are creatures they usally have the same power and toughness and the other abilities they have are usually very similar. There are a lot of different kinds of cycles.
This is a cycle across colors. These are done all the time across all sets. A good example of such a cycle would be the Eidolon cycle (e.g. Enigma Eidolon). They have the same casting cost, a sacrifice ability and the same power and toughness. These cycles are good for keeping balance in sets because they generally spread the power level evenly across the colors.
A vertical cycle is a cycle across all rarities. These cycles are usually less strict than horizontal cycles as the rarities all have different power requirements. A good example of a vertical cycle would be the red nightmare cycle from Torment (Spellgorger Barbarian, Soulgorger Orgg and Worldgorger Dragon). These cards all have a similar name, creature type and ability.
A loose cycle is a cycle of cards that have similarities, but are not too strictly following the pattern. A good example would be the big legend cycle from Ravnica. These cards are all legends with big stats that represent the leaders of their guilds but they are very different in mana cost, abilities and power and toughness.
These cycles try to keep as much of the cards the same as they can. An example of this would be the familiar cycle from Planeshift. These cards all have the same abilities and vary only in creature type, power and toughness, the cards are so similar that it is impossible to see them and not notice that they fit together. That is what a strict cycle is all about.