First you need to know hou many entries there are in the table (which I'll call 'quotes').
$query="SELECT COUNT(*) FROM quotes";
$result=mysql_query($query);
$total=mysql_result($result,0,0);
Now, let's make sure we have our random number generator warmed up:
srand((double)microtime()*1000000);
And generate a random number in the range 0..$total-1:
$random=rand(0,$total-1);
Now use use that in a SELECT statement's LIMIT clause:
$query="SELECT thisquote FROM quotes LIMIT $random,1";
And fetch the resulting item from the datbase the same way as before:
$result=mysql_query($query);
$randomquote=mysql_result($result,0,0);
This can be done much more sophisticatedly (perhaps using MySQL's RANDOM function in some way), but this is a good enough start.