Having become a half-human, half-ghost teenage superhero has significantly altered Danny's life and gained him great maturity. Danny started off the series as a naive and clumsy teenager who tries to find an identity for himself and his newfound powers. His early days had him yearning for popularity and often has him shallow and superficial.
Later in the series, Danny starts to move beyond the superficial to focus on his goal as a superhero, preferring to step out of the spotlight when he isn't on-duty. But he continues to uphold his vow to protecting his hometown and those in need. He has developed a more confident and mature attitude over the course of the series as he continues to develop his powers, but it is ultimately his loyalty to his cause that helps him to overcome the obstacles. Even when he is vilified by the public early in his career his sense of duty is so strong that he felt retirement was out of the question.
He said that his battles had a "pattern: "Witty banter, fight, kick-butt, lesson." However, he can be cocky in battle and at times underestimates his enemies due to his naïveté and young age, though he is just as often cautious. He is also often too emotional, which can be used as a disadvantageHe is sometimes overcome by his emotions and this shatters his confidence and self-esteem as well as his actions as a ghost hunter..
But Danny can also be responsible, considerate, and intelligent while battling. He tries to protect his loved ones. For instance, he tried to protect his parents from Vlad Masters, when he, unlike his family, knew Masters was half-ghost. He tries to protect his friends as well, since they may be more vulnerable to other ghosts since they are closer to Danny.  When he makes mistakes, he honestly tries to repair the results, with his friends and family.  
Upon receiving his ghost powers, Danny often had great difficulties controlling it. At the mercy to his teenage emotions in the beginning, his nervous behavior would often trigger episodes of intangibility or invisibility, much to his embarrassment. Besides his ability to switch between his human and ghost forms, his early powers composed primary of the most generic of ghostly powers; flight, intangibility, and invisibility. When he transforms, he shouts out "Goin' Ghost", (which Vlad Plasmius calls Danny's "Battle cry"), before two white rings intersect his body. When he flies, his legs often turn into a ghost "tail". He also gains super strength and enhanced abilities and agility his human form does not possess. He can manipulate his body and twist himself into odd shapes to avoid stray attacks. He is able to walk on walls, though this is rarely utilized.
He has a ghost sense that manifests as a blue wisp from his mouth that announces the presence of nearby ghosts. It is described as being akin to a cold breath during chilly weather and served as a precursor to his ice powers.
Danny gains a ghost ray early on which has become one of his most frequently used powers; becoming an energy ball that can expand in size and corresponding area of effect, and even be usable as a flashlight at low power. He possess a ghost shield that can protect him by deflecting attacks. He has the ability to overshadow people and ghosts, causing their eyes to glow green as well as change their voice to his. He later masters the power, able to cancel or control such obvious signs of possession. He is also capable of entering people's dreams.
One of his most powerful attacks is the Ghostly Wail, a sonic scream that can hit multiple targets. It appears to take massive amounts of energy to utilize, draining Danny heavily, so he often reverts to his human form after use. By the end of the series he appears to have mastered this ability (to an extent). He can duplicate himself, but struggles to master it throughout the series—often with disastrous, or amusing, results. He succeeds in splitting in two, then eventually four later on.
Danny is either able to breathe underwater when in ghost form, or likely does not need to breathe. It is unclear if he can breathe in space, though it is shown that he can handle its extreme conditions with little more than a space helmet. Danny can also later on produce an ice beam which can freeze the opponent among other things and so on. His ice powers, though, do not come in till later in the series since it takes a while for them to develop.