Shamatha is often translated as "calm abiding". It's the capacity in attention to abide with, hang out with, be in the experience of, whatever arises. Returning and resting in sensations and feelings again and again as attraction and aversion arises, we learn not to repress or act out. Eventually we see, we really know, that every experience is just an experience and we don't need to react. That seeing/knowing is vipashyana, insight.
Some practices and teachers emphasize the resting in experience; resting is good when you're habitually busy trying to understand or manipulate things.
Other practices or teachers emphasize the looking into specific experiences, for example the ones that cause us particular trouble; looking deeply is good when you're habitually passive.
Whether you emphasize resting in attention (shamatha) or looking deeply (insight), the two come together and happen simultaneously. We can rest, without distraction, in experience, and know its nature.
Everyone's got their own unique cluster of conditioning, but the flavors and particulars of one's emotions and behaviors don't really matter -- the practice is always about returning and resting in experience, so that you don't repress or act out so much, so that one can see/know the nature of experience, so that more choices are possible.
Buddhism is often thought or taught to be a problem-solving practice, but the practice is to rest in the problem, not try to make it go away. Things relax and become clear over time as you stop approaching experience as a problem. That's a paradox, but the profound things in life are. Easy problems are fixed with easy solutions, but Experience-And-How-to-Respond isn't easy. It gets easier as you cultivate a capacity in abiding.
People often give shamatha the once-over and try to jump into insight, but that's a mistake. Trying to analyze and problem-solve without the capacity to rest in experience is useless at best. At worst it's just a new version of the same old futile strategies that create the suffering we're trying to avoid.
You may benefit by devoting every other day's whole session to shamatha, so that insight practice doesn't slip into analysis, and digging into death, or karma, or reactive emotions doesn't stir things up beyond usefulness. As Ken McLeod says, deep capacity to return and rest in whatever's arising is what gives insight practice its power.
Here's one way to do it: on the alternate days, spend 15-20 minutes just returning to and resting in the breathing body.
Then for 5-10 minutes look into one particular emotion or behavior. Hold the question:
What is this? [craving, or aversion, or confusion, whatever you've chosen]
Touch, feel, taste every sensation and feeling that arises. When you get lost in a story, or you come up with "insights" or "answers," just gently return to holding the question What is this? and looking into the experience itself. As you hold the question, feel yourself breathing in and out.
After 2-5 minutes, give up all effort and just rest again in attention in the breathing body for 5-10 minutes. Feel your body sitting there, its weight, its shape, the space around it in the room. Just rest like that for a few minutes.