In my opinion, the conflict in this book is what Harry is doing, along with Professor Dumbledore, throughout the book -- looking for Horcruxes. The conflict is between Harry and Voldemort. Voldemort has hidden the Horcruxes years ago and Harry and Dumbledore have to find them now.
The climax of the book, to me, is when Snape kills Dumbledore during the fight up on the tower.
In a way, there is not a real resolution because the resolution is going to happen in the next book. After the climax, we see Dumbledore's funeral and we are left in doubt as to whether Hogwarts will open again the next year, but really nothing is resolved.
There are many conflicts in this book, ranging from Harry's quest to retrieve a very important memory from Professor Slughorn, to the love triangle between Ron, Hermione, and Lavender Brown. The most serious conflict would be Harry and Dumbledore's continued fight against Voldemort via investigation of Voldemort's past, especially Voldemort's knowledge of Horcruxes. An important gap in Dumbledore's records of the past is a memory that Professor Slughorn is unwilling to relinquish. Once Harry finally retreives the memory, the next task for Harry and Dumbledore is to actually find the Horcruxes and destroy them.
Another conflict would be Draco Malfoy's suspicious activities and the question of whether he is the cause of the various nearly-fatal accidents that have been occuring around school. Harry suspects throughout the book that Draco is a Death Eater, but is unable to find solid proof of this.
Harry is also very much put-off by Slughorn's attempts to include Harry in his "Slug Club", a group of students that Slughorn favors. And finally, several romantic conflicts arise between Harry, Ginny, and Dean, as well as Ron, Hermione, and Lavender. These conflicts are eventually resolved in various ways.
The more serious conflicts are resolved in the climax of the novel, a single night. Harry and Dumbledore successfully retrieve what they thought was a Horcrux, though it turns out to be fake. It is revealed that Draco is a Death Eater who had been plotting Dumbledore's assassination, and through his efforts many Death Eaters have entered the school and are attacking its students and teachers. A fight breaks out between the Death Eaters and the students and teachers, culminating in the death of Dumbledore and the relevation that Severus Snape is the Half-Blood Prince.
The resolution to the novel would probably be Harry's determination to find the Horcruxes, with Hermione and Ron's help.