user11995: There are obvious ones that I know, like queen and rook (or either) vs lone king, and I also know you can't checkmate with only a bishop or knight, for instance.  
But what if you have a rook and bishop vs a knight, is that possible?  What if it's vs a knight but with eight pawns?  I mean the list is endless.  In short, I just want to know when it's a good idea just to say it's a draw vs other times you just keep trying and trying to get a checkmate.  Thank you.

user924: The ones that are not possible are:

King + Bishop vs King
King + Knight vs King
King + Bishop vs King + Bishop if the bishops are on the same color square.

For the bishops, imagine black king on h1, black bishop on g1, white king on g3, white bishop on f3 giving mate. That isn't possible if the black bishop is also on a white square, say f1.
Otherwise, checkmate is possible with a given material balance. 
For instance, of course rook and bishop vs knight can lead to checkmate, because checkmate with a lone rook is possible, and it is possible that the rest is captured.
Any material balance that includes a pawn can lead to checkmate, because pawns can turn into queens.

11995: <p>There are obvious ones that I know, like queen and rook (or either) vs lone king, and I also know you can't checkmate with only a bishop or knight, for instance.  </p>

<p>But what if you have a rook and bishop vs a knight, is that possible?  What if it's vs a knight but with eight pawns?  I mean the list is endless.  In short, I just want to know when it's a good idea just to say it's a draw vs other times you just keep trying and trying to get a checkmate.  Thank you.</p>
 924: <p>The ones that are <em>not</em> possible are:</p>

<ol>
<li>King + Bishop vs King</li>
<li>King + Knight vs King</li>
<li>King + Bishop vs King + Bishop <em>if the bishops are on the same color square</em>.</li>
</ol>

<p>For the bishops, imagine black king on h1, black bishop on g1, white king on g3, white bishop on f3 giving mate. That isn't possible if the black bishop is also on a white square, say f1.</p>

<p>Otherwise, <strong>checkmate is possible</strong> with a given material balance. </p>

<p>For instance, of course rook and bishop vs knight can lead to checkmate, because checkmate with a lone rook is possible, and it is possible that the rest is captured.</p>

<p>Any material balance that includes a pawn can lead to checkmate, because pawns can turn into queens.</p>
