Almost every card in the game has an ability. These abilities override the rules, and allow you to do crazy powerful plays. To begin, look at the 4 pieces of each ability. For this example, we'll use the ability of Hedera.
During your main phase, act: A summon gains 3 strength.
At the beginning, in italics, is a condition. The condition must be met in order to use the ability. If there is no condition on the card, the ability is active at all times. (Unless it is a spell, in which case it is assumed to be "During your main phase")
Following the condition, in bold, is the cost. The cost must be paid at the start of the ability, before it actually gets used. Some cards do not have a cost on their abilities, which means you can use it as often as the condition allows for.
Following the condition and cost, there is a bold symbol. That symbol will either be "->" or ":". If there is a colon, it means the ability is optional. If the condition is met and you can pay the cost, you may use it. If there is an arrow, it means the ability is mandatory. If the condition is met and you can pay the cost, you must use it.
Finally, in normal text, is the actual ability.
In short, for Hedera, you take the card during your main phase, turn it on its side, and then a summon gains 3 strength.
Remember that you can use abilities as often as you want, as long as the condition is met and you can pay the cost. (IE: Inevitable Plague)
You may notice that some cards have little notes at the bottom of them that help explain what the ability does. These notes do not change the way the ability functions, but rather explains what the ability does. They are designed to help newer players get into the game and learn what their cards do from the first reading of them, or to help experienced players in niche situations.
Many abilities occur at the same time. If two or more abilities would occur at the same time, the player whose turn it is decides in what order the abilities occur. Mandatory abilities must occur first, followed by any optional abilities. Use the abilities one at a time, checking that the condition is still met and that the cost can still be paid for the next one. If they cannot, that ability is skipped in the list and you must move on to the next one.
As another example, if an ability requires multiple actions to be performed, do those one at a time. (IE: Perfect Infusion)
You can find information about certain card's abilities and how they interact with other cards on the Card List page.
There are certain phases in BOS that may require some explaining before you can start playing.
Opponent: Refers to a player who is attempting to defeat you. Does not count teammates.
Damage: Refers to the maximum amount of damage that can be dealt by a source. If a summon has 500 strength and battles something with 3 strength, the smaller summon will be issued 500 damage, but only take 3, because that is the maximum amount of damage it can take. Damage issued = 500. Damage taken = 3.
Control: If a card is on your side of the battlefield, even if it is owned by another player, it is under your control. Any card that is under your control that would be sent to a graveyard is sent to your graveyard, even if it is owned by another player.
Place a summon: Refers to placing a summon down onto the battlefield. Does not count as summoning.
This wizard's summon: Refers to the summon you control in front of that wizard.
Base strength: Refers to the boss or summon's printed strength.
Damaged boss/summon: If the boss or summon's strength is less than its base strength, it is treated as "damaged."
Defeat: A boss or summon enters a state called "defeated" before it is about to leave the battlefield and get sent to its controller's graveyard.
When it would be defeated: It is not in the "defeated" state yet.
When it is defeated: It is still on the battlefield.
After it is defeated: It is in a graveyard.