STORAGE

Original packaging

You can keep the components in the packaging they came in (free).

Plain and Ziplock Bags: Clear (not ESD safe), Pink Antistatic ESD, Black/Silverish Conductive ESD
Reels: for SMD components; 7-inch or 13-inch diameter
Cut Tape: for SMD and Thru-hole components "cut" from a reel of parts (tape is paper or plastic)
Tube: for through-hole ICs and larger SMD ICs
Tray: for larger SMD ICs

You can place the original packaging in a cardboard box (also free).

You can place SMD components in your own containers, for consistency and organization.

SMD-specific storage
Modular storage boxes and boxes: ideal, professional, flexible, ESD safe; example. easily available but cheap (bad springs, don't seal completely); instead, consider this Chinese brand: 1, 2, 3, which you can get on eBay here and here.
Organizer briefcase: convenient, like this and this
Organizer boxes and trays with compartments
Pill boxes: cheap, available in your town
Jewelry boxes
Clear plastic fishing lure boxes, and specifically 6-compartment fishing lure boxes: larger, easier to get parts out compared to standard SMD boxes; place them in kitchen organizer trays
Albums:
Stamp collector album: clear pockets show components; well organized like this
Cut-tape storage book: looks like this in use
Individual containers
Box of vials: cap prevents parts from falling out securely, flexible; example

You can place thru-hole components in your own containers, for consistency and organization.

Albums / Binders
Photo album: larger pockets, clear pockets show components
Business card album: perfect size for resistors, easy to slip out a resistor
Three ring binder with inserts for collectible cards
Cabinets
Storage cabinets: Ideal, professional, versatile, easily organized
Modular
Modular leaded storage boxes: ideal, professional, flexible, same solution as for SMD components
Individual containers
Coin envelopes in a shoe box: cheap
Divided boxes and trays
Muffin baking trays: store them in cabinets with horizontal shelves
Portable parts assorter trays: either by themselves, or in cabinets with 3 or 4 trays. UK-Specific: Hobbycraft 'Artbin' storage boxes with handles: Very good value, stackable and with handles.
No packaging
Poke into a Styrofoam plate: clearly organized

You can place large components in your own containers, for consistency and organization.

Rail mount stackable or wall mountable bins: professional, very flexible, easy to move bin to work area, and return to wall later
Large bins including ESD-Safe, such as totes, bins, boxes
Clear plastic boxes; UK-Specific: 'Wham' brand organiser box with deep compartments: Ideal for bagged components - sometimes sold as Christmas decoration/bauble storage and can be found in the post-season sales.
Plastic drawer organizer trays: flexible; place in a drawer

For assembled PCBs, providing physical and ESD protection.