For more than half the year I lived with a boyfriend who could not support himself, due to the fact he could not find a job. But we also live with his mother. She makes about $20,000 more per year than I do. She pays the rent, but we both equally supported him. Who should claim head of household? And who should claim my boyfriend (if he can be claimed at all)?

I figured that she has been used to getting a skim tax return, and has not claimed him in six years. And that since I make less than she does, I would get the maximum return. But I am not sure if I can claim head of household since she makes more than I do.

I am confused, if you can help, thank you in advance.

You can't claim your boyfriend since he did not live with you all year as a member of your household. So you can't be Head of Household.
His mother can claim this son even he is over 19 years under the requirements of Qualifying Relative as he is no longer her Qualifying Child:
1. The person cannot be your qualifying child or the qualifying child of anyone else.
2. The person either (a) must be related to you, or (b) must live with you all year as a member of your household (and your relationship must not violate local law).
3. The person's gross income for the year must be less than $3,400.
4. You must provide more than half of the person's total support for the year.

Here are the requirements for the Head of Household:
You may be able to file as head of household if you meet all the following requirements.
1. You are unmarried or "considered unmarried" on the last day of the year.
2. You paid more than half the cost of keeping up a home for the year.
3. A "qualifying person" lived with you in the home for more than half the year (except for temporary absences, such as school).

Qualifying Person for the Head of Household status.
1. Qualifying child who lived with you more than half the year.
2. A qualifying relative who is your mother or father.
3. A qualifying relative who lived with you for more than half the year and you can claim as exemption for him or her.

So check if his mother can claim Head of Household status.

MukatA's advice, as usual, is flawless.

I have nothing to add!