The core refers to the single strand of metal at the centre of the string, the material and shape of which determines the base tone and flexibility of the string. Historically, all cores were round, but modern strings typically use a hexagonal core as this leads to to a brighter tone. If you're seeking vintage warmth and booming low end, however, a round core is the way to go.
The winding is the wire wrapped around the core of lower strings (usually the 4th, 5th, and 6th). The type of winding determines the tone, texture, and flexibility of the strings. Higher strings (the 1st, 2nd, and 3rd) typically aren't wound.
Roundwound strings are the most common type sold today. They feature a round wire which is wrapped around the string core. They have a bright tone, a textured surface, and are generally the easiest winding type to fret and bend.
Flatwound strings, as you might guess, feature a flat wire wrap. They produce a dark and understated tone that suits clean, theoretical styles of music such as jazz. Flatwound strings eliminate finger noise but are stiff, and struggle to cut through the mix.
Halfwound or 'ground wound' strings are wrapped like roundwound, but then the outer layer is ground down for a smoother feel. Sonically, halfwound strings are somewhere between the brightness of roundwound and the warmth of flatwound with reduced finger noise.
You can view the full range of guitar strings available at Mannys here.