The major breakthrough in practical mathematics in the dark ages was the development of Hindu numerals -- the ones we use today. The Roman numerals used in the West were so complicated that is was practically impossible to do arithmetic. Leonardo de Piza brought the Hindu numbers to Europe shortly after A.D. 1200, but they were not universally acceptable for almost 500 years. The numerals were accepted right away by mathematicians because they simplified calculation. Bankers were very reluctant to accept Hindu numbers because the simplified numbers made forgery much easier.
Algebra was the great breakthrough in abstract mathematics during the dark ages. It was basically an Arabic invention, although it was greatly improved by Hindu mathematicians including Omar Khayyam -- a poet who was also a mathematician. Algebra introduced variables into mathematics and made possible a more concise terminology, and the solution of a whole new class of problems. At the end of the dark ages, the combination of algebra and geography would allow Rene Descartes to develop analytic geometry and the Cartesian coordinates -- developments that were an essential foundation for the development of calculus.
Trigonometry is also an Arabic invention that was developed during the dark ages. Trigonometry started as a set of techniques based on triangles -- hence the name -- but has developed into a branch of mathematics that has applications far beyond its origins. For example, the science of Fourier Transforms that was developed at the end of the dark ages is based on trigonometric functions. Fourier transforms today are used for the computer analysis of pictures and for encoding of information for satellite telemetry applications.
The first written reference to negative numbers was by the Indian mathematician Brahmagupta in the 7th century. Europe was resistant to the idea until the 18th century. The reluctance to the acceptance of negative numbers seems weird until we consider that European mathematical thinking was based on Greek geometry. In geometry there are no negative quantities. Many famous European mathematicians as late as the 17th century actively campaigned against negative numbers -- thinking that supporters of negative numbers were taking a wrong turn into nonsense. Up until the 19th century, negative numbers were described as "debt" and considered as different from "real" numbers.