Overall, in an addition reaction a pi bond is broken, and two new sigma bonds are formed.
The pi bond of a carbon-carbon double bond is a Lewis base (ie electron donor) and is used to form the first new sigma bond. The choice of calling it a Lewis base was very intentional as this means it can act like abase or as a nucleophile. So, which is it? Well it depends on the reagents present. If a strong acid is present the pi bond will act as a base and be used to form a sigma bond with a hydrogen through a proton transfer. If a good electrophile is present the pi bond will act like a nucleophile and be used to form a sigma bond with the electrophile through a nucleophilic attack. The result (except in the case of Br2) is the formation of a carbocation intermediate.
If the reaction involves an asymmetric alkene, two different carbocation intermediates are possible. In these cases, move the pi electrons to form the more stable carbocation. Remember carbocation stability is determined by the number of substituents and the ability to delocalize the positive charge through resonance. Also remember to be on the lookout for the possibility to form a more stable carbocation through a hydride or alkyl shift.
Carbocations are GOOD electrophiles, so a nucleophilic attack readily occurs to form the second sigma bond.
Want more visual explanations and specifics? Watch the videos listed below then practice using a carbon-carbon pi bond as a Lewis base in the puzzles listed.