Solid Wood is for larger budgets in particular climates. It is more expensive and due to wood reacting to temperature changes, often is not ideal.
Medium Density Fiberboard can be pretty awesome in a LOT of circumstances. It is cheaper than solid wood but won't warp with years of seasonal changes or humidity. If you are painting your cabinets then MDF is absolutely better than Solid Wood because there aren't any seems. MDF also cleans easier.
Hard Density Fiberboard uses more wood fiber than MDF. Laminate flooring is one place you can find HDF. Many people think that "well, if MDF is so great for cabinets, then HDF will be even better." That isn't true. HDF is great for flooring. No one is going to use cabinets the way a high traffic area of flooring gets used. You'll be paying more for a level of wear-with-all that is unnecessary 99% of the time.
Melamine is a very exciting cabinet material. And wall paneling. And Kitchen Island siding. And... the list goes on. Melamine is a type of very hard laminate that can be textured. It is modern without being boring.
For the tight budget Plywood fits the bill. It is also lighter than other types of materials and very easy to work with. Most amateur handymen can do a decent job working with various types of plywood, too. Plywood as your choice of cabinet material is just step 1. Plywood comes in various veneers and can look very, very nice.
A Veneer is a thin layer applied to another material, such as a thin solid oak veneer on plywood. Unfortunately it is totally legal to call something Solid Wood when it actually is just a thorough full surface covered veneer over a solid core center. Nearly everything is veneer these days. You can even do MDF and put a beautiful wood veneer on it.
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