I used to use a small cardboard box that I kept in the fridge. Then my daughter spilled water on it!
So I moved my seeds to a plastic tupperware with a lid to keep any liquids from spilling on it, thank you very much. 
Nowadays, I have four of those tupperwares in my fridge, because my seed stash outgrew the one. Here’s how each of those tupperwares is organized:
Sow in fall
(June through December)
Sow in winter
(January through March)
Sow in spring
(April through May)
Beans and corn to sow in spring
(April through May)
I kept forgetting to sow things that I had intended to sow at a particular time, so I finally decided organizing them based on when to sow them made the most sense. Now I can grab a tupperware based on the time of year it is and browse through it for ideas.
Meanwhile, the seeds are grouped together in large ziploc bags within each tupperware, each bag labeled based on the microclimates available in my growing space. These include:
Sow in fall box:
Needs stratification: dry shade
Needs stratification: moist shade
Needs stratification: partial shade (I put all “full sun” perennials that aren’t drought tolerant here)
Needs stratification: full sun (only perennials that are highly drought tolerant go here)
Winter sun
Winter shade
Sow in winter box:
Winter sun
Winter shade
Greenhouse: full sun
Greenhouse: moist shade
Greenhouse: dry winters (plants that will rot and die if they get lots of water in winter, so they have to spend winters in my greenhouse, whether or not they’re hardy to my zone)
Sow in spring box:
Beans: full sun
Beans: partial shade
Corn: partial shade
Squash: partial shade
Full sun (for everything else: if something can’t handle frosts and it isn’t a staple I’m willing to water regularly, it’s only realistically going to survive in my garden if it’s highly drought tolerant and can handle full sun)
I think that’s accurate, but I didn’t open them up to check, so I may have organized things a little differently, and I may reorganize them on a whim at any time.
For right now, anything that needs cold stratification goes in my “sow in fall” box. So do any winter crops that I’m trying to select for being able to germinate and survive the last half of our hot, dry summer growing season (which lasts through September) before flourishing as winter crops.
Anything that needs to be planted after the last frost goes into the relevant “sow in spring” box. (I have to use two boxes because squash, beans, and corn are all annuals with large seeds that take up a lot of space.)
Any new winter crops that I want to semi-coddle in order to get more seeds go into my “sow in winter” box. I have found it’s much easier to keep winter crops alive if I sow them in January or February, but they don’t get as big or produce as much food as the ones I sow in August or September, so I hope to be able to adapt all my winter crops to be happy to germinate during our (one and only) week of summer rain in August, and then sail through our coldest winter temperatures without even noticing. 
I used to separate things based on them being annual or perennial, but I eventually decided that distinction mattered very little to me — and it’s not even clear-cut with some species, such as Brassica oleracea, which can be annual, biennial, or perennial depending on that particular plant’s genes.
I also tried separating things based on plant family, but I also eventually decided that didn’t matter to me. Plant family didn’t tell me when or where I wanted to place things, after all. Especially with highly diverse families like the carrot family — I mean, skirret wants to be in moist shade, lovage and dill are both fine in full sun, and carrots seem to do best for me in partial shade. I eventually decided to only separate the annuals with very large seeds (corns, beans, and squash) into their own bags, and I only did that because it was a necessity for stuff to fit.
Eventually, I hit upon the idea of organizing things based on when and where to sow them, and I think that’s working very well. Then I can browse the relevant category for ideas to add into a polyculture that isn’t yet sufficiently severely overcrowded (
), and/or for ideas to fill a row that is currently sitting empty because I just finished removing the weeds and/or harvesting the crops from the previous season.
Your own microclimates will vary based on your own climate. For me, there is no category for “full sun with lots of water,” because nope. But for other people, there may be. Also note that my greenhouse has only three microclimates available: full sun, moist shade, and stuff that needs to be dry in winter but doesn’t really need extra warmth (so those can go in the coldest spots). When it comes to tropical and subtropical plants, I have specifically gone looking for tasty edible plants that are highly drought tolerant, and those are for the full sun canopy. I don’t have a partial shade option ’cause my greenhouse is small and therefore crowded: it’s either part of the canopy layer and therefore getting baked by our severe sunlight, or it’s not.
Eventually, I would like to downsize my seed collection because I have most of the perennials planted and growing merrily, so I only need to save the annual seeds for replanting. But . . . in practice . . . I want to landrace everything, so I’ll probably always have freshly-harvested seeds of perennials that I’m eager to find a place to squeeze in. 