Basically there are two different types of twins; identical (monozygotic - MZ) or fraternal (dizygotic - DZ)
Monozygotic twins develop when a single egg is fertilised by a single sperm and at some stage in the first two weeks the developing embryo splits in two, with the result that two, GENETICALLY IDENTICAL babies develop.
Dizygotic twins occur when two separate eggs are fertilised by two different sperm. These two fertilised eggs then develop independently. Dizygotic twins share the same type of genetic relationship as non-twin siblings, hence the term fraternal, meaning brotherly.
Very rarely, the two eggs are fertilised by sperm from different men, leading to cases where DZ twins can be half-brothers or sisters, sharing only their mother's genes.