Probiotic bacteria are "live microorganisms which when administered in adequate amounts confer a health benefit on the host." (WHO)
These helpful bacteria mostly come from just a few main families of bacteria (and yeasts) - the Lactobacillus, Bifidobacterium, Saccharomyces and Streptococcus families.
To truly be considered probiotic a bacteria shold be defined by strain as different strains from the same family do not have the same effect on health.
Presently, this often does not happen but, if possible, when chosing a probiotic supplement always try to choose one where they state what strain of probiotic good bacteria is in it.
For example, a bacteria such as Lactobacillus acidophilus NCFM comes from the Genus or family Lactobacillus, the species is acidophilus and the strain is NCFM.
Many manufacturers still consider this information "commercially sensitive" so it is not always available. If it is a well known brand with many positive reviews then it is probably safe to just accept that it is a helpful bacteria even if the strain is not stated.
This is not so important in foods where in naturally fermented food you will be taking in a wide range of good bacteria - some of them being probiotic and some of them being lactic acid bacteria - but all of them doing you good.
This is the ideal way to take your probiotic bacteria if it is at all possible.
Just as vitamins and minerals are essential to your health, so are probiotics. Get them through your diet if you can, but if not consider a good probiotic supplement.