The word of the LORD that came to Jeremiah: See 1.4. It is probably not necessary to identify him as the prophet here.
Elam was a country to the east of Babylonia, of which very little is known.
Zedekiah: See 1.3.
This verse can be rendered:
• This is the message the LORD gave Jeremiah concerning the country of Elam. This message came shortly after Zedekiah became king of Judah.
Quoted with permission from Newman, Barclay M. and Stine, Philip C. A Handbook on Jeremiah. (UBS Helps for Translators). New York: UBS, 2003. For this and other handbooks for translators see here .