What are the different methods for modeling something in blender? I know a few like mesh modeling and sculpting, but am interested in other methods to create shapes.
There really are only ~6 in blender, except for impracticable ones/not supported, the rest are derivatives:
1. Mesh modeling:
This is where you use edit mode, and move around, connect and disconnect vertices to form a shape.
There are a lot of sub-approaches, like extrude modeling and box modeling to name a few. As a rule, use extrude modeling if you're not going to have to eyeball proportions, since you can use the details from the start and work faster (like modeling off blueprints), and use box modeling for when your eyeballing stuff since it is much easier to adjust proportions.
Some people only like to do it one way, I'd try both and see what you like.
In this area Blender has very good tools.
This is where you work off of a very high poly mesh, and form a shape by pushing vertices in large numbers to form a shape, this can produce very creatve and organic workflow, generally used form non hard surface meshes, although some people sculpt hard surface meshes.
In this area, Blender has quite good tools.
3. Curve modeling
This is where a form is created using joined curves, spline modeling. This technique has been largely fased out, but was popular in the 90's (I think)
The most famous form is NURBS modeling, where you can create surfaces between curves, but as I said few people now use that method.
Blender does have curve support, but no support for trimming or viewport UV's.
4. Procedural Modeling:
This is where code/modifiers does the modeling work. A good example is the displacement modifier, where a mesh can be distorted based on a height map. It is easy to create rock surfaces using this technique.
Blender's tool set here aren't so advanced, though there are external add-ons that can help: eg: https://github.com/nortikin/sverchok
This is useful only for things like a fluid, liquid or gas/ it also is useful for modeling damage or clothing.
The fluid and to some degree the smoke simulator are limited, but the cloth simulator is pretty well featured
These are basically shapes that are enclosed in what can best be descried as a shrink wrap shell... They merge together, making it easy to modeling something like a peanut. When they are separated sufficiency they split and go back to their original form. There are few practical applications I know of for meta balls in blender.
See: http://en.wikipedia.org/wiki/Metaballs (this example only shows a few but you can have 100's)
Other Methods of Modeling:
These mostly include scanning and photo-modeling, but Blender doesn't include tools to do this out of the box, you'd need specialized software for this.
Besides Mesh (Polygon) modelling in Blender we have support for most of the now standard techniques.
In no particular order, and non exhaustive
Sculpting aims to be a digital implementation of the workflow involved in the traditional Art of clay sculpture (start with a virtual ball of clay and push + pull + pinch + lay strips + carve + flatten + polish, etc). Made popular mostly by ZBrush, but has a fair implementation in Blender. This way of modelling has drawn a wide audience including non-traditionally trained artists, and can be used to generate both organic (soft-body) and mechanical (hard-body) shapes. A web search for the term "Blender Sculpting dyntopo" returns many examples to give you an impression of how far it can take you.
Similar to the Set theory uses geometric algorithms to produce Union / Intersection / Difference of 2 selected Objects. In Blender this relies on the Boolean Modifier, and both meshes must be manifold (and not self intersecting). It allows for fast Carving of one shape from another, or joining two shapes together, or even extracting just the parts of two meshes that are in the same 3d space. Because the current implementation is Polygon/Mesh based the resulting mesh will often not be ideal for further manual editing where you might expect topology to be directly related to the contours and features.
Patch Modelling (Curves and Nurbs Surfaces)
With these forms of modelling the surface itself is not defined vertex by vertex by us, but by control points, knots, weights and interpolations. These surfaces remain defined purely in mathematical terms until you convert them to Meshes. Blender distinguishes between 2D and 3D Curves. While 2D curves are great for logos / typography and profile shapes, the 3D switch can take one Curve and use it as a profile along a trajectory. These techniques allow for non-destructive experimentation without dealing directly with the resulting mesh for as long as you want.
Parametric node based Modelling
At the moment (mid 2015) the PyNodes system Sverchok is the only Blender add-on that's explicitly aimed at generating geometry and has over 170 operator nodes. Sverchok takes its inspiration from Grasshopper for Rhino. Targetting architects and designers it assumes some knowledge of basic Vector and Matrix maths, familiarity with Linear Algebra isn't required but helps. If the names Zaha Hadid or Greg Lynn resonate you might want to investigate parametric modelling.
Similar in a sense to ZSpheres (if you know what those are), it allows you to define a basic edge based mesh and assign radii to each vertex, then it grows a 3d mesh around the structure. You will see this in a lot of Blender sculpting movies as a starting point.