Summoning sickness(oft abbreviated to simply "sickness") is a rule/condition that applies to creature spells in Magic: The Gathering. When a creature spell is cast, they are inflicted with sickness. In lore terms, the creature is disoriented from being summoned, and must take time to get its bearings before engaging in combat. The consequences in literal terms as they apply to the game are that a creature cannot attack or use activated abilities which require the creature to tap or untap until its controller's next upkeep. If control and/or ownership of the creature should change before sickness wears off, sickness will wear off on the upkeep of the player gaining control of the creature. Once sickness has worn off for a given creature, it can no longer be re-afflicted by sickness unless it is moved from the battlefield to a different zone and back again.
Sickness applies exclusively to creatures, and does not inherently affect other permanents (artifacts, lands, enchantments). Though for game balance, there are instances of artifacts lands or enchantments that enter the battlefield tapped.
What sickness most effectively means for game balance is that most creatures cannot be dropped on a player's main phase and swung immediately, though there are strategies and deck compositions that actively work around this. The mechanic that navigates around sickness is haste - which simply means the creature is unaffected by sickness. This might seem pithy and small as mechanics go, being as it can only make a difference for the turn in which the creature entered the battlefield. But in competitive play where speed of play is much more intense, haste can make a big difference. It can be a highly convenient way to catch opponents off guard and throw quick damage or use its abilities before there is an opportunity to respond.
Summoning sickness directly affects the viability and advantage of the flash mechanic, which allows a permanent to be cast at instant speed. In effect, players that have creatures with flash (or if they have Leyline of Anticipation or something) can keep their mana pools open for counterplay until the last end step before their turn. Once priority has been passed to them on said end step, they can use the rest of their mana by casting creatures right before their untap step and upkeep, upon which summoning sickness for the cast creature will immediately wear off. This is a general mentality of blue decks on a basic level - giving yourself room for counterplay and responses while denying the same to your opponents.
Sickness is technically an informal term, though has been used in print before. The "idea" of haste has existed since Alpha Edition, with text on some creature cards reading "(creaturename) can attack the turn it comes into play." However, starting with Mirage from the Mirage block and lasting all the way through the end of the Urza block, the antiquated pre-haste text read "(creaturename) is unaffected by summoning sickness." In the Summer of 1999 6th edition was released, bringing about a massive slew of modifications to cards, rules, and game mechanics. Amongst these changes was the introduction of the term haste ("This creature may attack and tap the turn it comes under your control").
MTGsalvation wiki article: Summoning Sickness
Good tidings of great joy and thanks to Hazelnut for helping to clarify haste's origins.