Stability:
The general stability order of simple alkyl carbocations is: (most stable) 3^o > 2^o > 1^o > 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 utilized 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. Alternatively, loss of H+ can generate a p bond.  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 if it generates a more stable carbocation

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