Transform is a keyword action that appears on the all of the double-faced cards and Moonmist. Transform is not a keyword ability, as the word itself does not signify the entire ability; rather, it is a keyword action like cast, exchange and exile.
When a creature transforms, it turns over. Its base characteristics are those of the side now showing. Any existing effects that were applied to the other side will still be applied to this transformed card. For example, if a Gatstaf Shepherd was targeted by Titanic Growth, it would be a 5/5 creature until the end of turn. If it were to transform, it would be a 6/6 Gatstaf Howler. Similarly, all equipment, enchantments and counters will remain on an object when it transforms and will continue to apply to that object.
It does not happen often, but it is possible for an object to transform, and then transform right back. For example, if the controller of a Ludevic’s Test Subject with four counters on it were to activate its ability and then without passing priority activate the ability again, it will transform into Ludevic’s Abomination when the first abilities resolves and then back into Ludevic’s Test Subject.
Cards (or tokens) that are not double-faced cannot transform. Any time one attempts to transform, nothing happens. This also applies when a card is a copy of a double-faced card–it is still a single-faced card and therefore unable to transform.