For tumblr, a gif must be under 1MB, and the size dimensions must be under 500x500 pixels. (you also may have to experiment with the width sizes of your gifs some, because some widths will freeze out a gif and it won't move, but I've found my 350 pixel x any height; 300 pixel x any height, and 245 pixel x any height gifs show up properly)
Getting a nice quality gif under 1MB takes a lot of experimenting and sometimes a little bit of luck, but here's how I generally go about it:
1. First you want to resize your gif. As mentioned above, anything under 500x500 pixels is acceptable, but not every width will work. Obviously, the smaller the dimensions of the gif, the easier it'll be to get it under the 1MB limit, but it'll also show up more grainy on tumblr when forced to show at large dimensions in a photoset. I generally go with 350 pixel width x by any height gifs.
Even at these dimensions, this gif is still above the size limit when there are no adjustments made to it. You can check the size of your gif in the bottom left hand corner of the 'Save for Web & Devices' pop-up screen.
2. One of the easiest ways to decrease the size of a gif (other than deleting a number of frames; the less frames you have, obviously, the smaller your gif will be -- a lot of making gifs is deciding what part of an action is most important to be seen, and downsizing the animation to that small moment of action) is to adjust the amount of colors used in the gif. This option is in the top, right hand corner of the 'Save for Web & Devices' pop-up screen. You can either change your gif to one of the set amounts (256, 128, 64) or type in an amount of your choice. My example gif decreases to 878.5K (under the size limit!) at 128 colors, but there will be a noticeable difference in quality at the lower color amounts (especially if you have a gif with a lot of different colors; the thing I notice most is the color of lips/etc washing out into a brown or pale color instead of showing up pink/red).
You can also adjust other options for your gif in the other drop-down bars of the 'Save for Web & Devices' screen. Changing the 'Selective' option to 'Perceptual' or 'Adaptive' will change the color distribution within the gif and can also affect the size; these are things you can experiment and tweak to get the desired look you want, as well as find ways to make minimal adjustments for staying under the 1K size limit.
3. Another extraordinary helpful tool for gif-making is the Adjustments option. If you don't see a tab for this on the right side of your photoshop screen, you can find it under Window -> Adjustments at the top of your screen. Adjustments are extremely helpful when making gifs because they affect all the layers below them, so you don't have to add your color editing/etc to every single frame of your gif individually. And, if you want to make sure your adjustments will affect every frame (important when lowering the opacity of an adjustment layer, which will be discussed in step 5!), select your first frame then press shift + right click on your last frame, and all your frames should be selected. (Every selected frame will be affected by your edits; you can deselect some frames or only select one frame if for whatever reason you only want an edit to affect a certain amount of frames, etc)
4. We're going to first look at the 'selective coloring' option of adjustments, which is something I commonly use to adjust the size of my gifs. Because the number of different colors is directly proportionate to how large a gif ends up being (as shown above where lowering the number of colors used in a gif also lowers the size!) adjusting a gif to have less colors/more similar coloring throughout can make it small enough to be under the size limit. On your selective coloring adjustment tab, you'll see a drop-down menu that lets you choose what color you'll be adjusting (it starts on Red). I generally adjust the Blacks and Neutrals, as they affect a larger portion of the gif's coloring.
Here's the sample gif with the Blacks tweaked (using the sliders; you can experiment with these as well) -- I've found that making the Blacks of a gif blue-tinted (less yellow in this example) and darker (more black) can quickly help lower the size of a gif, but you'll need to play around with each specific gif to see what works for it. A general rule of thumb: try to get the colors more similar throughout a gif, if there's already a lot of blues and greens go for a blue tint, if there's a lot of reds try to make the entire gif have a red tinge to the colors.
5. Another helpful edit I make (but a more complicated one; it helps if you're familiar with working with photoshop) on gifs is to adjust the contrast, but I generally do this through a few steps instead of just using the Brightness/Contrast adjustment layer.
To return to the original Adjustment panel and add another adjustment layer, you want to hit the little arrow on the left hand corner of the Adjustment panel; in the screenshot I have an arrow pointing to it with 'Return to Adjustment Layer.' Then, you'll want to create a Vibrance adjustment layer by selecting the Vibrance option. The reason I do this is because, to get contrast on my gif, I change my layer from a 'Normal Layer' (this will be shown in the Layers panel below your Adjustments panel; it's the underlined 'Soft Light' option in this screenshot) to a 'Soft Light' layer, which does a more fine-tuned contrast than simply selecting a Brightness/Contrast adjustment layer. However, changing a layer to soft light also tends to play up the colors in the gif/make them too strong (yellows get REALLY yellow, etc) so by doing this on a Vibrance adjustment layer, you can lessen the vibrance (the underlined '100' in the screenshot) or the Saturation until your gif is back to the shades and amount of color you want. You can also change the intensity of the contrast by adjusting the opacity of this layer (shown next to the 'Soft Light' option underlined in this screenshot).
You can really use any adjustment layers you want/play around as much as you want (adjustments layers are a great way to make a black and white gif, as well) but these are the two that I use most often, so I wanted to explain them in the most detail. The best thing about adjustment layers is that if you don't like the result, you can simply delete that adjustment layer without affecting the original gif image layers at all!
You can keep checking the size of your gif in the 'Save for Web & Devices' screen throughout each adjustment you make, and continue tweaking the amount of colors/etc, until finally you end up with a gif you're satisfied with that's under the size requirements for tumblr, like so!
And here is the original, over-the-tumblr-size-limit gif:
And here is the finished result, which at 994.2K is under the 1K size limit!
Despite the fact that there are less colors used in the second gif and it's a smaller file size, there's no particularly noticeable change in the quality of the gif; just in the coloring, which is my general goal when making gifs.
Hopefully this is helpful to you all! Feel free to ask questions, I'll do my best to answer them. And note that there's a hundred helpful tutorials on gif-making out there, and some by people who are much better acquainted with making gifs on tumblr than I am, so please seek them out as well!