Kicker is a keyword ability that has been around for awhile and made several return trips to Magic expansions. It, like many other keyword abilities, allows your cards to be more flexible and do different things at different times depending on what you pay.
A kicker ability is basically an extra cost that you can pay when you cast a spell. The cost you pay is listed after the word kicker. There will also be lines of text stating what happens if the kicker cost is paid. For example; If Goblin Bushwhacker is kicked when it enters the battlefield creature you control will get +1/+0 and haste until the end of turn.
You decide to pay the kicker cost and make the payment when the spell is cast. You can not announce a spell and then when for the spell to resolve in an attempt to sneak the kicker in. Additionally, kicker abilities can only be paid once. However there is a variant of kicker called multikicker that does allow you to pay the kicker cost as many times as you want. Apex Hawks is an example of a card with multikicker.
Finally, kicker is an optional ability, you do not have to pay the cost if you do not want to. Some kicker abilities cause a targeted triggered ability to go on the stack or has an additional aspect of resolution that targets an object. If a spell is not kicked, these abilities will not trigger or need to have a target chosen in any way. For example, if a Gatekeeper of Malakir is not kicked, then its ability will not trigger and you will not choose a target player.