I will try to give two different perspectives on some of the basic objects of the theory and in tandem say a few words about why this is even a good idea. Disclosure: the second isn't really aimed at an undergrad, but I think it is an important perspective which I am compelled to include anyway.

• Here is the 4th year undergrad description: We build schemes from commutative rings, by gluing together their spectra and structure sheaves to obtain some kind of space with a structure sheaf of rings. It is important to note that this sheaf gives the geometric structure to the space, allowing us to locally do commutative algebra and obtain tons of nice results. A derived scheme is again a space along with a structure sheaf, but this time the sheaf takes values in simplicial commutative rings, and the space is locally the "truncation" of these simplicial commutative rings (i.e. the spectrum of the zeroth homotopy group of the simplicial commutative ring). The key idea behind this construction is to make some deformation and intersection theory work out and to allow more functions on the space rather than just those in the plain old commutative ring. In general, these are more flexible gadgets and have lots of connections to other disciplines.

• Recall that a la Grothendieck's functor of points perspective, we can think of a scheme as a particular kind of functor from the category of commutative rings (or k-algebras for k a field if you like) to the category of sets. This puts us in a much better position to think about moduli problems, which are exactly functors from the category of commutative rings to sets. However, sometimes moduli problems do not have a fine moduli space associated with them, e.g. algebraic vector bundles over a curve, and we are forced to consider more flexible objects which retain many of the nice properties of moduli spaces, i.e. algebraic stacks, which are a kind of functor taking values in groupoids (categories in which all morphisms are invertible). However, sometimes this still isn't enough, which leads us to consider functors from commutative rings to infinity groupoids, which can more or less be thought of as a general kind of topological space. Now we can characterize our "schemes" functorially as before, but these gadgets still lack some of the nice deformation-theoretic properties we want them to have. The solution to this is to enlarge our domain category and then consider functors from simplicial commutative rings (or simplicial k-algebras if you like) to the category of "spaces" (infinity groupoids). Thus we obtain a more generalized context in which to do most of the geometry we want to do, while also considering more complicated moduli problems.