Accordingly means in accord with, or in correspondence with something. If you learn that your so-called best friend is spreading nasty rumors about you, act accordingly and stop being her friend.
Accordingly is the adverb form of accord, which means harmony, or agreement. We often use the phrase "act accordingly," which just means to behave in a logical manner based on what you feel, learn, or are told. If you are half-starved after being lost in the woods for three days, when you come across a house with a full kitchen, you will act accordingly. Accordingly can also mean "therefore." You were hungry, accordingly you ate.