First off: if you are Ashkenazi, at least a few of those who appear in "2nd-3rd cousins," especially those near the bottom of that list, are unlikely to be your real cousins. See Jennifer Mendelson's article for a cogent explanation why: https://medium.com/@CleverTitleTK/no-you-dont-really-have-7-900-4th-cousins-some-dna-basics-for-those-with-jewish-heritage-857f873399ff

(It is possible, though much less likely, that someone who has a stronger match and shows up in the "1st-2nd cousins" section will also not be your genuine cousin. The rest of my post is premised on the assumption that they probably are indeed your genuine cousin.)

There are a number of ways to figure out how names in your AncestryDNA list who you don't recognize are related to you. Here are a few:

1. Contact the person directly, using the "Message" button in AncestryDNA, and ask them.* Not everyone will respond, but many will!

2. Ask around. See if your known relatives who you are in touch with have heard of this person before. 

3. Do traditional research like you would for any other name you don't recognize; see if you can identify who this person is, where they came from, who their family members are, etc. Even if the person didn't link their results to a tree in Ancestry, you can often figure out quite a bit based on their name alone.

4. Use the Shared Matches tab on their Ancestry profile to narrow things down. If you already know that 6 people on your mother's side of your family match you on AncestryDNA and 4 people on your father's side match you on AncestryDNA, how many of those people match this mystery person whom you don't recognize? If they match all of your known paternal cousins while appearing to match none of your known maternal cousins, that is a pretty decent hint that this person is likeliest to be another paternal cousin of yours. 

5. Convince more of your relatives to also take AncestryDNA tests to see how they match the mystery person. Expensive, but might be worth it to figure out if you really have a long-lost close cousin!

*If your family is close, and you would know who all your first and second cousins are (yet can't figure out the connection to this person despite AncestryDNA saying they are your first cousin), it's probably a good idea to proceed gently when communicating with this mystery person. There is always a possibility that one of your aunts, uncles, or close cousins has a secret, like giving up a baby for adoption or having a child out of wedlock, and that even the child has no idea that one or both of the people who raised them are not their biological parents.

Good luck!

Adam Turner