You can claim a boyfriend or girlfriend and their children as dependents if they are your qualifying relatives.
Boyfriend/Girlfriend: To be a qualifying relative (dependent) you don't have to be an actual relative - however,
whoever claims them must provide more than 1/2 of the individuals support,
they must not earn more than $3,950 in 2014 (social security doesn't count),
they must not file a joint return with another
If not an actual relative, they must live with the person claiming them all 365 days of the year
they must be a US citizen or resident of the US, Canada or Mexico
they are not a qualifying child of another taxpayer.
Boyfriend/Girlfriend's Children: In addition to the above requirements, to claim your boy/girlfriend's children who are not also your children, they must meet all of the above requirements and:
your boy/girlfriend must not be required to file a return,
he/she does not file a return claiming the children, and
they meet all of the requirements above to be a qualifying relative.
An additional issue - a boyfriend/girlfriend and their child (who is not also your child) will not qualify you for Head of Household because you are not related. Also, the child will not qualify you for earned income credit, child tax credit or the child and dependent care credit (again, because you're not related.)