They close the gap. Plenty of settings sit low or spread wider than the shank, so a straight band bumps into the setting and leaves a sliver of space under the rest of the ring. Some people like that look. Others feel it makes two rings that do not quite belong together, and a loose pair tends to twist and rub.
A fitted band follows that outline instead. The two rings nest, the top of the hand reads as one design, and your diamond stays the star.
The names overlap, but the shapes differ. A curved band arcs gently in the middle, a soft dip that suits round and oval heads. A contour band usually traces the setting's outline more closely. A notched band keeps a mostly straight line with a small cutout, the divot people search for, which the setting drops into.
Any of these can be plain metal or lined with small diamonds. A diamond curve wraps extra sparkle around your main diamond, almost like a partial halo; bare metal keeps the pair quieter and is quicker to clean.
Look at the profile of your engagement ring first. Designs where the head sits low over the finger, or where a halo setting spreads wider than the band, are the most likely to leave a gap beside a straight ring. Large center stones and elongated shapes set close to the finger often do the same.
Raised designs can go either way. In cathedral settings, metal arches lift the center stone above the shank, which often leaves room for a straight band to slide underneath. A hidden halo adds a ring of small diamonds below the main stone that shows from the side, so study the ring in profile before you decide.
View your engagement ring from the side and from above. Notice the shape of the space under the head: a wide, shallow arc points to a curved band, a narrow drop to a notched one. Then compare metal color and width so the two rings match where they meet.
Order both rings in the same size, or very close to it, so the curve lines up with the head every time you put them on; the free ring sizer from Rare Carat lets you find that size at home. A curve that sits even slightly off center will tilt the pair and undo the point of buying it.