Tapped and untapped are just two physical states a card can be in — turned sideways, or standing upright. Untapped means ready to use; tapped means already used, and it stays that way until your next untap step.
Summoning sickness is a separate but related idea: a creature that just entered the battlefield can't attack or use any ability with a tap symbol in its cost until it's been under your control since the start of your most recent turn. It can still block, though — sickness only stops attacking and tap abilities, nothing else.
New players almost always assume a freshly cast creature can jump straight into the fight. It can defend itself immediately — it just can't throw the first punch.
— the most common turn-one-creature mistake
Tapping and untapping are defined in rule 701.26; summoning sickness comes from rule 302.6.
Comprehensive Rules · Rule 701.26a–b
"To tap a permanent, turn it sideways from an upright position. Only untapped permanents can be tapped. To untap a permanent, rotate it back to the upright position from a sideways position. Only tapped permanents can be untapped."
Comprehensive Rules · Rule 302.6
"A creature's activated ability with the tap symbol or the untap symbol in its activation cost can't be activated unless the creature has been under its controller's control continuously since their most recent turn began. A creature can't attack unless it has been under its controller's control continuously since their most recent turn began."
"Summoning sickness" itself is informal shorthand players use for this rule — it doesn't appear as a defined term in the rules text, just the underlying restriction.
How to take advantage of it
Haste skips the wait entirely. Creatures with haste ignore summoning sickness completely — they can attack or use tap abilities the same turn they enter, which is why haste is prized on both threats and mana dorks.
Vigilance keeps your attackers untapped. Normally attacking taps a creature, leaving it unable to block. Vigilance means it stays untapped after attacking, so you get offense and defense from the same creature.
Tapping down blockers is real removal. Effects that tap an opponent's creature during your turn don't destroy anything, but an already-tapped creature can't be declared as a blocker — functionally clearing a path for your attack.
Watch what's untapped on the opponent's side. Untapped mana or an untapped creature is information — it tells you what they might still be able to do before their next untap step.