Twins, triplets or quads are generally the result of multiple ovulation by the ewe. Some ewes produce more than one egg each time they come into heat, and if these make it to term the result is twins, triplets or more. Some breeds are more "prolific" (produce more offspring per ewe) than others, and this trait can be selected and improved by keeping the offspring of ewes that produce twins or triplets.
What role does the ram play? As we've said before, the ram provides 1/2 of the genetics to each lamb. If the ram has twinning in his pedigree, that is if his mother produced many sets of twins or triplets, his father's mother produced twins or triplets, and so on, then chances are very high that his female offspring will also produce twins or triplets. Since rams don't ovulate (produce eggs), we can't measure directly their twinning capability like we can in the ewes where we can see and count how many lambs they produced. So in rams we have to look at their female relatives to make inferences about their genetics for twinning.
If you want to produce twins, you need to keep the female offspring from ewes and rams that have twinning in their pedigrees, and in the case of the ewe that she produced twins regularly.
Another good indicator of twinning potential is early fertility. If a ewe produces her first lamb or lambs when she is a yearling, she is demonstrating early fertility. Many of these will go on to produce twins as 2 or 3 year olds and older. If a ram is born as a single to a ewe lamb, look at her mother to see if she has twinning in her performance history. Also ask about the ram's sire's mother to see if she has twinning in her records.