Welcome to the forum.

In C8 of Sheet 2, use this:
Formula: copy to clipboard 
=IFERROR(INDEX(Sheet1!$A$2:$A$5,MATCH(B8,Sheet1!$D$2:$D$5,0)),"nothing due")

This looks up the date on Sheet 2 (B8) in the list of dates on Sheet 1 (D2 to D5 - expand it as you need), then returns the matching result from column A on Sheet 1 (A2 to A5 - again, expand it as you need). If there's nothing on that date, it would normally return the error #N/A, so the IFERROR part of the formula instead tells it to return 'nothing due' - change that to whatever you want (putting "" will leave the cell blank).
A word of warning - if you have multiple items due on the same date, this will only return the first one. It is possible to return them all, but it's quite a bit more complicated - if you need this, get back to us.

Hope that helps.