You need someone's public key to accomplish two things: encrypt a message destined to this person, or verify the signature of a message coming from this person.
You might obtain the public key as an ASC file from the person himself, either as an email attachment or by hand. In this case save the file on your machine, then from Key Management select File → Import keys from file. Choose the ASC file you just saved. Enigmail will add this public key to your keyring.
Another way to find someone's public key is to download it from a keyserver.
Select Keyserver → Search for keys and insert as the search term a part of the name or email address of this person. You may also search for key IDs. The keyserver will return a list of public keys that match. Tick the checkboxes on the left of any key you would like to import and Enigmail automatically will do it for you.
Later, you can also refresh your public keyring in part or as a whole by the menu commands Keyserver → Refresh Selected Public Keys and Keyserver → Refresh All Public Keys.
You might also find a public key published as raw text, for instance in a personal web page. It will appear like this:-----BEGIN PGP PUBLIC KEY BLOCK-----
Version: GnuPG v1.4.9 (MingW32)
mQGiBEgN3cERBAC+2os2hoACip4EiKxEzv+iVHOWaznOJIGZY9zY4y8C0BhUP++q
ccgO9vgNO1vIXApYvJctwX9HFJieNdlsBrWOR69hPBAbDo+3BbOKwJFYgq8akYnv
tBCOdCNfOFwQs/8XdOH25/Oig+UjKhgxwKjkdd1UCj7shdGioXOvjO13xwCgzrGa
k2oA9Bne3hW+jUPjJlU4UbkD/i7mbQfFwTgxcXfRfsVDnkmPc+QvKe00ajfRP31o
(... 30 lines omitted ...)
3bRLiE8EGBECAA8FAkgN3cECGwwFCQlmAYAACgkQW9ZLGa9iatRGlACeIyyIBTGj
uhrZRubL8gSdI3HyPzEAnAqWD1g+DvEHyEOZ0/2rUcsj7CT2
=3RyK
-----END PGP PUBLIC KEY BLOCK-----
This is also what you see if you open a public key ASC file with a text editor.
Select the whole block of text, including the lines -----BEGIN PGP PUBLIC KEY BLOCK----- and -----END PGP PUBLIC KEY BLOCK----- and copy it to the clipboard (Ctrl+C under Windows). Then choose Edit → Import keys from Clipboard to import this public key into your keyring.
You can search and add public keys to your keyring at any time. Enigmail will also offer to do so automatically when you receive a signed message from someone and you do not have his public key; we'll see this in Section 8.2.3. If you followed all instructions up to here, by now your keyring should contain your own key pair and a number of public keys purporting to other people.