A Cost is an action that must be performed in order to activate a Card or Effect, or to keep a Card on the Field. This is distinct from activation conditions, which are requirements that you have to meet in order to activate a Card or Effect.
As the name suggests, Costs usually involve you making a sacrifice of some type, such as paying Life Points (for example, "Solemn Warning" requires you to pay 2000 Life Points); discarding, tributing, banishing, detaching or sending a Card(s) to the Graveyard (For Example, "Glow Up Bulb" requires you to send the top Card of your Deck to the Graveyard).
Fulfill the Requirement
If a Cost cannot fully be paid (such as if an active Effect prevents the Cost from being paid or you cannot fulfill the requirements for the Cost), none of the Cost can be paid and the Effect cannot be activated. For example, if Cost specifies that you must send a Card to the Graveyard while "Dimensional Fissure" is on the Field, you cannot activate the Effect.
Interaction with other Cards
Because Costs are distinct from Effects, if you move a Card to a different location as a Cost, Cards and Effects that can only be activated when the Card is moved by a Card Effect cannot be activated. For example, the Effect of "Shaddoll Dragon" cannot be activated if it is sent to the Graveyard as a Cost to activate "Phoenix Wing Wind Blast". Likewise, even if a Card's Effect is being negated (such as by "Lost Wind"), it is still possible to activate its Effects (which would require paying the Cost).
An activation Cost is a mandatory action that must be done in order to activate a Card or Effect. This is separated from the rest of the Effect by a semicolon (or without PSCT by the word "to", a comma or the word "then"). It must be paid at the beginning of the Effect's activation. The Cost itself is not activated, cannot be negated, and occurs at activation time, not resolution time.
When an Effect features a Cost, if the timing is correct (up to the colon), the Cost is paid and any targeting is done (up to the semicolon for Cards with PSCT, and normally up to the word "to" / comma / "then" for Cards without PSCT), then the Card is placed on the Chain (or starts a Chain if there is not a current Chain). After either the rest of the Chain resolves or both players decide not to Chain to the activation, the Card or Effect resolves.
If a Cost starts with "You can..." the Effect is optional and the Cost is only paid if you choose to activate the Effect. Even if the Effect is mandatory, the Effect cannot be activated if the Cost cannot be paid, and no Cost is paid if you can't pay all of it.
Cost distinct from Effect
A Cost is not an Effect. As such, the Cost is paid before the Effect takes place, regardless of whether or not the Effect is negated. If a Monster's Effect is negated as long as it's on the Field, and its Cost requires it to leave the Field, the Monster can activate its Effect. For example, if "Exiled Force" activates its Effect while affected by "Lost Wind", because it is not on the Field when it resolves (due to Tributing itself as a Cost), its Effect is not negated by "Lost Wind".
Costs from Continuous Effects
Cards such as "Chain Energy" have Continuous Effects that impose a Cost on performing certain optional actions, such as declaring an attack or Summoning Monsters. If you cannot pay the Cost for the Continuous Effect, you cannot perform the action.
A Maintenance Cost is a Cost required to keep a Card on the Field. These Costs are typically paid during your Standby or End Phase.