Costs, discounts, and optional additional costs
A card's cost is built in layers, in a fixed order, and most disputes come from applying a discount, an "ignore" or an extra payment in the wrong layer. Learn the order and the rest follows.
Worked example. Stalking Wolf costs 4 Energy and 1 Order Power, and says "As an additional cost to play me, kill a Bird, Cat, Dog, or Poro you control". That is a mandatory additional cost, written with "as an additional cost" and no "may" (rule 356.2.a.1). With no such unit to kill you cannot play it at all. Compare Akshan, Mischievous, who says "You may pay two Body Power as an additional cost": that one is optional. Whatever you decide about the optional cost, the total is worked out in order: base cost, then additional costs, then increases, then discounts, then changes to the total (rule 356.1, rule 356.2, rule 356.3, rule 356.4 and rule 356.5).
Kinds of cost. A cost can be a Base Cost, an Additional Cost, a Cost within Instructions ("do X to do Y") or an Applied Cost (rule 204). Costs must be paid to get the linked effect (rule 203). A cost is still considered paid when the game action behind it is replaced (rule 203.2 and rule 357.2.a). If the action is impossible, the cost cannot be paid and the linked effect does not happen (rule 203.3). Written "pay X to do Y", as in Power Nexus's "you may pay four rainbow Power to score 1 point", the payment is a cost within an instruction, so it still counts as paid when it is replaced (rule 204.3 and rule 203.2). Only an instruction to pay resources with no linked effect is not a cost: in "you may pay 4. If you do, ...", a replaced payment means the later instruction does not execute (rule 205). Paying a resource can be declined, which undoes the play of a card or ends a non-card cost (rule 445.2).
Ignoring costs. "Ignoring its cost" sets the base Energy and Power costs to zero (rule 356.1.b.1). "Ignoring its Energy cost" zeroes only the Energy and the Power still applies (rule 356.1.b.2). Later additional costs can raise the total again (rule 356.1.b.3). "Ignoring any and all costs" sets the total to zero, non-standard costs included (rule 356.5.a).
Discounts. A discount that applies to one component of the cost is applied when that component is added and before other discounts (rule 356.4.c). Discounts to the total come after those (rule 356.4.d). The player chooses the order within each group (rule 356.4.c.1 and rule 356.4.d.1). A minimum in a discount applies only to that discount (rule 356.4.e), and Energy and Power costs never go below 0 (rule 356.6). Discounts can reduce additional costs, even to 0, and an optional additional cost counts as paid if the player decided to pay it, whatever they actually paid (rule 356.4.f and rule 356.4.f.1).
Optional additional costs. Accelerate and Repeat are optional additional costs (rule 805.1 and rule 820.1). Accelerate costs one Energy and one Power matching the unit's domain, and paying it creates a delayed replacement effect: the unit enters ready instead of entering exhausted and then becoming ready, so abilities about units becoming ready never see it (rule 805.1.b.1, rule 805.6 and rule 805.6.a). With Repeat, the instructions are executed once more for each Repeat paid, but the card is played only once (rule 820.3 and rule 820.3.a). The cost of Deflect is a mandatory additional cost for whoever chooses the unit (rule 356.2.a.2). Effects that need a card's cost use its printed or copied cost, whatever was actually paid (rule 206).
Common mistake. "Ignoring its cost means I pay nothing." Only the base costs are zeroed: Accelerate or a Deflect tax still has to be paid.