The empire was at the center of interactions between the Eastern and Western worlds for six centuries. With Constantinople as its capital city,and vast control of lands around the eastern Mediterranean, the Ottoman Empire was, in many respects, an Islamic successor to the Eastern Roman (Byzantine) Empire.
Ottomans are most remembered as pioneers of gunpowder infantry and artillery (NOT inventors). Also they were famous for adopting the roman legion idea and building the infamous Janissary Corps, which were by far well trained, and experienced than counterparts. (til the decline,when they started acting like praetorian guards)
Mostly their achievements were militaristic. They successfully invaded and took over major parts in south-east Europe. But their most memorable achievements is the downfall of the western roman empire.
Also accredited with the idea of marching band music, well trained professional soldiers, their cultural impact-paintings,music etc etc.
The empire went to decline largely due to the inability of the new rulers, (none as good as their predecessors). The final blow came at WW1, when the empire dissolved to make home for Turkey.....phew i'm done.
oh they also made the first super-guns btw....