The difference is in which maps are admitted. In increasing order of specialization (and in modern advanced, not elementary high school terminology), topology is the geometry where maps are only required to be continuous, differential geometry allows only maps which are "smooth" (usually C^infinity), analytic geometry allows only maps defined locally by convergent power series, and algebraic geometry is the most restrictive of all in allowing only maps which are locally defined by polynomials or sometimes rational functions.
In the same way the objects studied in each geometry are often taken to be locally the zero locus of functions of the appropriate kind, at least in the last three geometries: a differentiable manifold is locally the zero locus of a smooth function (with maximal rank derivative), an analytic variety is locally the zero locus of a power series; an algebraic variety is locally the zero locus of a polynomial map.
Isomorphisms between two objects in each category are defined in the analogous ways: existence of mutually inverse maps of the appropriate kind. Since the categories above require increasingly more special maps, algebraically isomorphic objects are also analytically isomorphic, differentiably isomorphic and homeomorhic. Analytically isomorphic objects are also differentiably isomorphic and homeomorphic but not necessarily algebraically isomorphic.
It is of interest to aks when one can reverse these implications, i.e. to know when one can reduce the question of isomorphism to a more general, less restrictive, category.
A big theorem implies that in the case of projective algebraic varieties analytic isomorphism does imply algebraic isomorphism. Other big theorems imply that certain differentiably isomorphic algebraic varieties have certain algebraic invariants also equal. In dimension two at least, topologically equivalent, i.e. homeomorphic surfaces are also differentiably isomorphic.
The "index" versions of the Riemann Roch theorems say that certain analytic invariants of algebraic varieties, such as the "euler characteristic" (alternating sum of dimensions of algebraic or analytic sheaf cohomology groups) of certain line bundles or vector bundles, is actually a topological invariant. E.g. for a line bundle L on a Riemann surface, the difference h^0(L) -h^1(L) of the ranks of the algebraic sheaf cohomology groups equals d + 1-g, where d is the degree and g is the topological genus, a purely topological invariant of L.
generalizations of these concepts also exist, but these are the most basic meanings of these words.
the geometry taught in high school, or what used to be taught, is Euclidean plane geometry, a special case of differential geometry. In differential geometry one studies notions involving measure and curvature, and a special case is the study of surfaces whose curvature is constant everywhere. Euclidean geometry is that most special case where the curvature is not only constant everywhere but equal to zero everywhere. I.e. high school geometry is the geometry of a flat 2 dimensional plane.
So called classical non - Euclidean geometry is usually the study of surfaces with curvature constantly equal to some negative number everywhere. Spherical geometry is the study of a surface of constant positive curvature.
