Do I claim my 25 year old who is living at home as a dependant

Did 25 year old earn more than $4050 in 2016?

That depends on whether the person meets both the general requirements to be a dependent and meets the test to be a qualifying relative. Here are the complete rules:
Rules for Claiming an Exemption for a Dependent:

- You cannot claim any dependents if you, or your spouse if filing jointly, could be claimed as a dependent by another taxpayer.
- You cannot claim a married person who files a joint return as a dependent unless that joint return is only a claim for refund and there would be no tax liability for either spouse on separate returns.
- You cannot claim a person as a dependent unless that person is a U.S. citizen, U.S. resident alien, U.S. national, or a resident of Canada or Mexico.
- You cannot claim a person as a dependent unless that person is your qualifying child or qualifying relative.

Test to be a Qualifying Child:
1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year and younger than you (or your spouse, if filing jointly), (b) under age 24 at the end of the year, a full-time student and younger than you (or your spouse, if filing jointly), (c) any age if permanently and totally disabled.
3. The child must have lived with you for more than half of the year (except for temporary absences such as for school)
4. The child must not have provided more than half of his or her support for the year.
5. The child is not filing a joint return for the year (unless that
return is filed only as a claim for refund).
6. If the child meets the rules to be a qualifying child of more
than one person, you must be the person entitled to claim
the child as a qualifying child.

Test to be a Qualifying Relative:

1. The person cannot be your qualifying child or the qualifying child of any other taxpayer.
2. The person either (a) be related to your in one of the following ways:
Your child, stepchild, foster child, or a descendant of any of them
Your brother, sister, half brother, half sister, stepbrother, or stepsister.
Your father, mother, grandparent, or other direct ancestor, but not foster parent.
Your stepfather or stepmother.
A son or daughter of your brother or sister.
A brother or sister of your father or mother.
Your son-in-law, daughter-in-law, father-in-law, mother-in-law, brother-in-law, or sister-in-law.
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 $4,050 (social security does not count).
4. You must provide more than half of the person's total support for the year.

There is a very good worksheet to help you determine how much support you provide. It is on page 16 of IRS Pub. 501
https://www.irs.gov/pub/irs-pdf/p501.pdf