There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit and student status test, a relationship test and a residence test. Only a QC qualifies a taxpayer for the Earned Income Credit (EIC) and the Child Tax Credit. The Other dependent (qualifying relative) credit (ODC) is worth (up to) $500 per dependent and is non-refundable. That is, it can only be used to reduce an actual tax liability.
Since none of you are related to him, his only chance of claiming you and the children is under the qualifying relative rules, which have already been listed above. He can only get the (up to) $500 ODC credit, not the child credit or EIC.
To claim your girlfriend's children, they must meet all of the Qualifying relative rules above and:
--- your girlfriend must not be required to file a tax return,
--- she does not file a return claiming the children
