Mother cultures are active, live cultures that are awake and ready to be added to your milk or dairy to begin fermentation right away. They are typically LAB (lactic acid bacteria) although they can be a mix of primary acidifiers and adjunct cultures which help with flavor and texture more than they produce lactic acid.
DVI cultures (direct vat inoculation) are freeze-fried cultures which are shipped and stored frozen and have a little bit of powered medium, often lactose or powdered milk. They must be rehydrated and then mixed into the milk. DVI cultures require some extra time to “wake up” and switch to their fully-functioning state and begin multiplying in earnest to ferment the milk.
Typically, DVI cultures come in flat bags with a limited amount of culture, and can be expensive for home cheese makers to purchase. They must stay frozen and sealed well, but getting exposed to extra moisture nor oxygen. In deep freezer conditions such as -18 F many kinds of DVI cultures will last a long time, potentially years, although they may will lose strength over time.
Mother cultures, often made from DVI cultures, are essentially milk that has been populated with a known-good culture, allowed to acidify and form a loose yogurt consistency (much like clabber from raw milk), and then that resulting yogurt is stored in the fridge (for use within a week or so) or in the freezer (for use later on).
A mother culture, simply allowed to warm up to somewhat near room temperature, has no lag phase like DVI cultures do. When it is added to the milk it will begin working and multiplying and acidifying right away. The initial ripening step for the milk, where a pH drop of 0.05-0.1 is expected, can be much shorter and even instant when using a mother culture.
If you have a DVI culture, making a mother culture from it is generally straightforward. In a clean, sanitized container warm up milk to a typical temperature for the culture (e.g. 85-95 F or so for mesophilic cultures, and 100-110 F or so for thermophilic cultures). It is preferable to use pasteurized milk to grow the cultures in so that you keep only the strains you are interested in. Once warmed, add the manufacturer recommended amount of DVI culture to the milk, and keep anything from getting into it to contaminate it. Try to maintain its ideal temperature, but if it falls to room temperature it will take a little longer but is unlikely to cause any other issues. Wait for it to loosely coagulate from the lactic acid production.
If you are using another type of culture, for example another mother-like culture such as clabber, whey from a previous cheese making session, etc then calculate what 3% of the volume of your milk is, add that much cultured medium, and let it coagulate as above.
Once you have coagulation, check it out to make sure it isn't obviously gassy nor chunky, and smells good (cheesy or yogurty). If all went well, either store your new mother culture or in the fridge, reasonably sealed, or freeze it for future use. Many cheese makers will ladle it into ice cube trays and freeze it that way (and pop out the frozen culture cubes into a bag and put it back in the freezer) so they have a known amount of culture per cube that they can easily grab later and thaw for a cheese making session.
In the fridge the mother culture should easily last a week before is probably a good idea to repeat the process to strengthen the culture. In the freezer the culture will last for months.
You can make a mother culture out of any DVI culture (or other culture), so long as at least one of the bacteria strains in it is a LAB which is a primary acidifier (it focuses on producing lactic acid). Adjunct-only cultures are not good candidates for mother cultures, so consider combining them with a primary acidifier to make a mother culture.
Also, some cultures are primarily mesophilic and so need to be cultured at a temperature below 103 F. Some are primarily thermophilic, and benefit from culturing at that higher temperature (104-108 F). Some cultures are mixed, for example the farmstead cultures such as MA4002, and if you culture these at mesophilic temperatures the thermophilic components will lag behind and become out of balance. In these cases, it is recommended to make separate mesophilic and thermophilic cultures, and mix the two later right before you make your cheese in the proportion that you want.
Normally you would use between 2-3% of the milk volume as mother culture, mixed into your warmed milk once the mother culture is thawed. Use the full amount for pasteurized milk, and possibly less if you are adding it to raw milk. Also, some cheese making recipes call for more or less culture in order to manage the rate at which the milk and curds acidify, and you can scale the amount of mother culture you use similarly to slow down or speed up acidification of the milk or curds.