Why is an HIV infection considered "incurable"?
The reasons why HIV is "incurable" (a misnomer) are legion:


HIV is a retrovirus, which means it inserts its own genome into the host cell's genome.  You must therefore kill each and every infected cell to rid the body of the virus.
HIV is a lentivirus, which means it has a long incubation period, so it can "lay low" before symptoms are readily detected.
HIV infects CD4+ helper T-cells, macrophages, and dendritic cells, which are responsible for mediating the host immune response.  Thus, it infects specifically the cells we need to fight an infection.
HIV drastically reduces the number of CD4+ T-cells.
HIV has a number of viral proteins that prevent some of the body's antiviral mechanisms
HIV can infect via cell-free (large number of particles, low infection rate) or cell-to-cell (low number of particles, high infection rate) routes.
HIV is wildly variable.  It has a small RNA genome (about 10kb) that mutates very rapidly.  Given the number of viral particles produced each day during infection — well over 1 billion — every single base is mutated every day.


I'm simplifying matters a bit — you should just read the en.Wikipedia article for your own research — but that's a rundown of some of the reasons why.