we get a triangle
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It is a section formed by a plane at right angles to the axis of the cone.
it a cone cut in half from the top
A cylinder, a doughnut with a slice cut out by radial planes, a sphere with two slices removed (possibly in a wedge shape), a cone with its top and bottom cut off by horizontal planes, a paraboloid with top and bottom cut off by horizontal planes.
You have a 3-d shape. If you slice of a part of it with a very very sharp instrument (a sharp sword or a laser, for example) the cross section is the shape of the newly cut surface. So, if you start with a sphere, any section will be a circle. Of different sizes depending on how high or low you cut, but they'll all be circles. If you have a cone, a dead vertical cut, through its apex (top point) will be a triangle. But as you move off the apex, you will get a parabola. Cut is horizontally, though, and you will get circles - ncreasing in size as you move further from the apex. Cut it at a slight slant and you will get ellipses (ovals). And so on.
It is the section of a cone between two parallel planes that are perpendicular to the axis of the cone. In simpler terms, it is what is left of a solid cone when the pointed end is cut off.