The existence of Devil Fruits is deeply intertwined with One Piece lore. It's one of the major elements constituting the series' power structure, as well as one of the manga's major hooks. The series has long explained the different kinds of Devil Fruits, as well as the powers they grant to whoever is lucky enough to consume them. However, only after the time skip were they further explored, specifically the Devil Fruit awakening.
With Devil Fruits being a major source of power in One Piece, a good chunk of its thousands of characters possesses a Devil Fruit ability, especially within the higher echelon. These fruits and their powers are classified into three types: Logia, Paramecia and Zoan. The three types each offer a different kind of ability. These abilities are able to grow stronger through the use of a Devil Fruit awakening.
Devil Fruit awakening became a main point of interest during Luffy's fight with Doflamingo. The latter was the first to exhibit said power to an intriguing extent, and he also explained to Luffy that only Devil Fruit users can tap into this ability. However, the specific method of activating it remains unknown. One conjecture is the user needs to meet a certain power level while also comprehending their ability to a certain extent. Law and Kid also noted that awakening their Devil Fruit powers took a huge toll on their bodies, thus limiting the duration they can use it.
While awakening had only been expounded on during the "Dressrosa" Arc, it's first shown during Luffy's Impel Down prison break. Four of Impel Down's Jailer Beasts are known to be awakened Zoan Devil Fruit Users, giving them enhanced strength, speed, and durability. They also stay in their animal forms with human attributes, and they act on pure instincts. However, it is not clear if this is due to their forced awakening, as Crocodile noted, or a general trait of awakening a Zoan Devil Fruit.
When Doflamingo awakened, he was able to transmute his environment into controllable strings. He could also manipulate these strings without the need of being in contact with it. Katakuri's Devil Fruit awakening also manifested in similar ways. During his fight with Luffy, he was able to make the walls and floors of the Mirror World act like mochi.
Both Law and Kid unveiled their awakening during their fight with Big Mom. Law's awakening allowed him to cover objects with his space-distorting skill, which further gave him the ability to operate on his opponents from the inside. Meanwhile, Kid's awakening let him magnetize objects and other living beings. He typically used this in the presence of a large amount of metal to maximize the damage. However, his ability required him to touch his target prior to magnetizing them. Unfortunately, the awakening of Logia Devil Fruit powers is yet to be revealed -- assuming that they can also be awakened.
One Piece has only started to touch on Devil Fruit awakening, and there's still only a handful of characters who have demonstrated the said ability. But given that the series is nearing its biggest war, it's reasonable to expect One Piece to provide more details about awakening in the near future. Perhaps Luffy may even awaken his Gum-Gum Fruit.