Conceptually, addition is the reverse of elimination
 

What does the term "electrophilic addition" imply ? A electrophile, E+, is an electron poor species that will react with an electron rich species (the C=C) An addition implies that two systems combine to a single entity.

Depending on the relative timing of these events, slightly different mechanisms are possible:

• Reaction of the C=C with E⁺ to give a carbocation (or another cationic intermediate) that then reacts with a Nu^-
• Simultaneous formation of the two s bonds

• Recognise the electrophile present in the reagent combination
• The electrophile adds first to the alkene, dictating the regioselectivity.
• If the reaction proceeds via a planar carbocation, the reaction is not stereoselective
• If the two new s bonds form at the same time from the same species, then syn addition is observed
• If the two new s bonds form at different times from different species, then anti addition is observed

Stability:
 

The general stability order of simple alkyl carbocations is: (most stable) 3o > 2o > 1o > methyl (least stable)

This is because alkyl groups are weakly electron donating due to hyperconjugation and inductive effects. Resonance effects can further stabilize carbocations when present.

Structure:
 

Alkyl carbocations are sp2 hybridized, planar systems at the cationic C center.  The p-orbital that is not utilised in the hybrids is empty and is often shown bearing the positive charge since it represents the orbital available to accept electrons.

Reactivity:

As they have an incomplete octet, carbocations are excellent electrophiles and react readily with nucleophiles (substitution). Alternatively, loss of H+ can generate a p bond (elimination). The electrostatic potential diagrams clearly show the cationic center in blue, this is where the nucleophile will attack.

Rearrangements:
Carbocations are prone to rearrangement via 1,2-hyride or 1,2-alkyl shifts provided it generates a more stable carbocation. For example:
 

Notice that the "predicted" product is only formed in 3% yield, and that products with a different skeleton dominate.  The reaction proceeds via protonation to give the better leaving group which departs to give the 2o carbocation shown.  A methyl group rapidly migrates taking its bonding electrons along, giving a new skeleton and a more stable 3o carbocation which can then lose H+ to give the more stable alkene as the major product.

2o carbocation to 3o carbocation

This is an example of a 1,2-alkyl shift.  The numbers indicate that the alkyl group moves to an adjacent position.
Similar migrations of H atoms, 1,2-hydride shifts are also known.

Reactions involving carbocations:
1. Substitutions via the S_N1
2. Eliminations via the E1
3. Additions to alkenes and alkynes (HX, H₃O⁺)