In the simplest sense, plane geometry is generally associated to two dimensions and solid geometry to three dimensions. In perception all the things we use and/or see in our mundane lives are in one way or another related to solid geometry that is they all have 3 dimensions that include length, breadth and thickness. In reality there is nothing in this world that can exist only in 2 dimensions because we live in a 3D world, even a rectangular piece of paper does have all the 3 dimensions. The only difference is that the thickness of the paper is so small that we cannot perceive it on a relative scale. But if you imagine us as tiny insects or ants, than at that level the thickness of that paper is truly huge and it does exist in 3D. Therefore plane geometry represents 2D shapes like circles, lines, triangles, rectangles and many other shapes and solid geometry is concerned with 3D solids, spheres, lines in 3 spaces and anything and everything that has 3 dimensions.
In theory we can say that an object having solid geometry can be deemed into an object having plane geometry when any one of the dimensions of the solid object approaches zero, or in other words is extremely small on a relative scale. One daily life example we can easily realize are photographs, they are mere reflections of 3D objects on a plane as one of the dimensions is missing in the photographic scale. So for ease of understanding a human is solid geometry and its photograph can be understood as plane geometry. One more simple exercise that can help drive the point is using a piece of paper. The piece of paper with infinitesimal thickness lying flat on the ground can be understood as plane geometry, but as soon as you pick it up at an angle it is solid geometry because now the same plane of paper has to be represented in three dimensions. Now after picking up the paper if you roll it into a hollow cylinder all the 3 dimensions of the paper come alive and it is now truly solid geometry.
Plane geometry is used more for ease of understanding of properties of certain geometries before we can truly appreciate things in solid geometry. Therefore all our earlier geometry courses are based on deriving and learning equations for geometrical shapes like circle, square, triangle and so on. Once we thoroughly understand the concepts of plane geometry it becomes much simpler to extend our thinking into 3 dimensions by expanding the same equations we mastered for plane geometry on solid geometry.