Programs: Photoshop and GIMP. GIMP will be covered second as the programs can do this independently and I'm not as proficient with GIMP. I just think the gif layout for GIMP is better than Photoshop's so I use both when making gif avatars/signatures.
Paint is optional.
Operating system: Windows 7
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Let's compare an image I extracted, png vs gif format.
png | gif |
 |  |
There's barely any difference, but I would say the most prominent parts are the top of their heads and the extreme left and right where their hair is. This is because gif does not allow partial transparency, it either has the pixel there or throws it away completely. Because of this, some pngs look fine but become weird when turned into gifs.
First, we need the color of the background where the image will be placed on. This is why this method 'locks' the image, different forums have different backgrounds.
Get a screenshot and use the color picker to get the background you want. If you have a program that saves it instantly to your computer where you can load it into photoshop/GIMP immediately, that's great. However, I'll be doing the traditional print screen method and then pasting into paint for those who don't know how to/don't have the program.
Now we'll compare png, gif with stroke and gif.
There should be no difference between png and gif with stroke but the same differences between gif with stroke and normal gif compared to png and normal gif.
GIMP later, I have to eat now.
Edit:
Comparing the three of them again, this time the gif with stroke from GIMP.
Now you might say, "What if I want something like your current signature? The stroke looks weird when intersecting with the animation."
When deleting the extra part from the frames, turn off the stroke before using it as your deleter. After that, turn it back on and merge down the first frame of animation which will overlap with the stroke.
Example (lazy to get the same gif, using random one.
):
I know, the strap could use more cleaning.