Probiotics are the good bacteria that live in your digestive tract, and they serve a lot of useful purposes. If their population gets too low they have trouble keeping their claim on your system, and bad forms of bacteria and yeasts take over.
Along with a balanced diet low in processed sugars, taking a probiotic regularly helps keep the good bacteria population up which supports a healthy immune system. A probiotic supplement doesn’t need to be taken daily unless you’ve had a course of antibiotics or a severe imbalance. There’s no standard recommendation, but if you take a supplement try having one every week or two and notice if your digestive system seems to stay healthy.
Most people think of yogurt when they think of getting probiotics, and there are some excellent non-dairy yogurts that are made with properly cultured probiotics. One thing to watch out for is that many yogurts (especially the dairy ones) need to be pasteurized, which kills the delicate probiotic bacteria. Look on the package to see whether the yogurt has been pasteurized or not.
There are also probiotic supplements that are wonderful for getting a more concentrated source of probiotics. Remember, probiotics are bacteria which feed and grow on sugars, and there’s no need for any animal products in their life cycle. So there are plenty of great vegan probiotic supplements out there.