What do the terms 'Human Emancipation' and 'Political Emancipation' mean? And which one of these are promoted by capitalism?
Marx discusses human and political emancipation in a (relatively) little known work, On the Jewish Question. There are many copies online for free.
Marx characterizes society under bourgeois as a place where rulers take their private concerns and turn them into public concerns (effectively pushing their personal preferences to the public, who cannot refuse them).
Political emancipation is a state of being where people are freed from this process.
For example, under the French ancien regime people were forced to accept the religious preferences of the state (Roman catholicism). This is a state without political emancipation. Contrast this with the 21st century America. The state does not overtly prefer any religion and citizens have freedom of (and from) religion.
This is important because it provides the bourgeoisie the ability to rule in their own economic interest without having to satisfy other social concerns. Although Marx doesn't use the term, it's easy to think of it as being similar to rationalization
These lecture notes from Florida International University provide this quote:
the modern State emancipates the Jews, not by freeing them from the domination of religion but by freeing itself from religion, by giving recognition to no religion and hence putting the Jews on an equal footing with everyone else
The Problem with Political Emancipation
Here is the problem Marx gets to: political emancipation frees the state, but it is a barrier to the freedom of individual citizens. Marx is a materialist, and as part of his project he claims that religion is something people cling to when their economic/material lives are unsatisfactory [Read more @ SEP]. When the rulers have freed themselves of religion, they are less constrained in oppressing workers. However, under political emancipation the public is still divided by religion, preventing them from either perceiving their oppression or resisting it.
The solution is the concept of human emancipation. Although Marx never really defines human emancipation, the way he contrasts it with political emancipation suggests that it is the actual emancipation of humans, not the state, from oppression.
Using religion as an example again, political emancipation from religion means that the state is now unconstrained by religion. However, citizens may still be religious and (in Marx's terms) are alienated from their labor by religion. Human emancipation from religion would indicate that citizens are no longer alienated from their labor by religion.