Thanks to the range of bathroom sizes across different homes, there are quite a few options for bathroom storage solutions out there. Depending on how much space you have available (and how much storage you wish to for) different options would work for you.
Want to make use of the space under the sink? => Vanity
A vanity base is simply a cabinet below your sink, concealing the piping and giving foundation for the sink to rest on. A spacious bathroom vanity is a must for any space efficient bathroom. While not strictly a bathroom cabinet, a vanity is a handy storage solution for the toiletries you need around your sink itself.
When choosing a vanity, the most important factor is to ensure that it matches the overall theme of your bathroom decoration, and does not look out of place from your other cabinets and shelves.
Wish for a sturdy storage solution? => Frameless Cabinets
Frameless cabinets are basically storage boxes. Usually constructed out of durable materials like plyboard or synthetic polymer sheets, frameless cabinets are one of the sturdiest bathroom storage solutions out there.
These units have a solid top as well as a back panel, which makes them suitable for both wall mounted installation or a floor plinth.
Most such cabinets have holes drilled at 32-mm increments of height along each side panel. These allow the cabinet to be fitted with hardware like European hinges, cabinet joinery fittings, shelf pins or even drawer slides.
Looking for something slimmer? => Face-frame Cabinets
The problem with frameless cabinets is the space requirements. For some bathrooms with irregular walls, it can be difficult to mount a high density box properly.
That is where face-frame cabinets come in. These cabinets generally lack a top panel, as well as possessing a partial (or very thin) back. The focus is on a sturdy hardwood (or synthetic resin) front that gives a very rigid 'frame' to the whole cabinet, and acts as a strong base for hinges.
Because of this unique construction, face-frame bathroom cabinets come in slim units that can fit in any bathroom or any wall with minimal customization. What is more, the strength of the frame means that you can go easy on the materials for the side panels, driving down the cost for the whole thing. As a result, face-frame bathroom cabinets form the bulk of storage in any modern bathroom.
Require a more spacious box? => Freestanding Cabinet
For some styles (like antique) the more popular design options for bathroom cabinets just do not fit. What you need for these styles is a more traditional 'furniture' look.
Freestanding cabinets are exactly for this purpose. Designed like a console vanity or a chest of drawers, these cabinets have their own legs. This removes the need for installation, as you just need to place the cabinets in your bathroom.
These cabinets also tend to have much more space for storage, whether you go with a tall or squat form factor. They can be very useful for imparting a 'living room' feel to your bathroom.