These are words, or a set of words, to describe a pre-defined ability.
For example, this card has a Keyword ability;

Flying:

A creature with flying can’t be blocked except by creatures with flying and/or reach. A creature with flying can block a creature with or without flying.

There are many other keyword abilities, all are described here.

An activated ability is an ability that can be activated by a player by paying a cost. The cost can be almost anything, such as sacrificing a permanent, paying mana, paying life, or tapping the permanent with the ability .
Activated abilities are usually found on permanents, but can sometimes be used on cards in other zones, such as the graveyard or a player's hand.
Only an object's controller (or owner, if it has no controller) can activate its activated abilities unless the ability says otherwise.
Also, you can usually activate an activated ability any time you have priority, unless it says otherwise.
Except for those that are mana abilities, activated abilities use the stack.
A player cannot activate an activated ability of a creature that includes the tap symbol or the untap symbol in its cost unless he or she has controlled the creature continuously since the start of his or her most recent turn.
This condition is informally known as "summoning sickness".
Activated abilities cannot be countered by spells or abilities that counter spells, because they aren't spells.
However, there are cards, such as Stifle, Squelch, and Voidslime, that can be used to counter these abilities.

Most activated abilities are written in the form "[Cost]: [Effect]". An exception to this are loyalty abilities of planeswalkers which are also activated abilities and which use a different notation.

A triggered ability is an ability that automatically does something when a certain event occurs or a set of conditions is met (the latter is called a state-triggered ability).

A triggered ability can be identified by the words "when," "whenever," or "at", which will usually be found at the start of the ability.
The phrase that contains one of those words lists the conditions (the trigger event or trigger state) where the ability will trigger.
When it does, the ability goes on the stack (unless it's a mana ability), and responses can be played.
The standard format for a triggered ability will be [Trigger Condition],[effect].

The example card has 2 triggered abilities, one triggers when it enters the battlefield, the other triggers when it dies.

A static ability is an ability of an object that is always "on", and cannot be turned "off". Keyword abilities are also static abilities.
The example card has three static abilities;
1. Vigilance
2. Other creatures you controll get +2/+2
3. creatures your opponents control get -2/-2.

Static abilities apply only when the object they appear on is in play unless the card specifies otherwise or the ability could only logically function in some other zone.
Static abilities are always 'on' and do not use the stack.
The ability takes effect as soon as the card enters the appropriate zone and only stops working when the card leaves that zone (or it says it does).

A mana ability is an activated ability that creates mana (such as with Llanowar Elves, Birds of Paradise, and Ur-Golem's Eye), or a triggered ability that triggers off of mana creation and makes more mana itself (such as with Overgrowth or the last ability on Gauntlet of Power).

Mana abilities are among the few activated or triggered abilities that don't use the stack or require passing priority to resolve, so they cannot be responded to or targeted.
As soon as the ability goes off, the mana is created.
In practice, this simply means that a player can use mana abilities any time he or she pleases, usually while playing a spell or ability and paying the cost.
Other effects of the mana ability, such as the damage that sometimes happens when using Adarkar Wastes, for example, also occur immediately and can't be responded to.

Spells that generate mana, such as Seething Song, use the stack normally, and can be responded to and targeted.