How do I make my textures transparent instead of white?

Technically, this is a graphical design question, but I'll give you the run down.

We learn at school that all colours are made up of three primary colours. Red, green and blue. Computers often read colour as red, green and blue (RGB) or red, green, blue and alpha (RGBA). Alpha is the quality that gives transparency.

Paint offers no way to manipulate the alpha. You can set a "background colour", and within the scope of paint, you can create the illusion of transparency, but it can not physically save an image as having transparency in it.

More advanced image editing software can generally handle transparency fairly well. Photoshop is the most popular one. The software can be fairly cheap, now, if you buy a dumbed down version as a CD key. GIMP is less user-friendly, but is just as powerful as Photoshop. And better yet, GIMP is free.

Keep in mind that when you save your textures, there are certain file formats that do not support transparency. If you are unsure, save your textures as .gif or .png for animated and non-animated textures, respectively.

In short: Microsoft Paint doesn't support transparency. As Timelord64 said, you can use GIMP to create textures with transparency. When you create a new image, make sure to go over to the layer manager (default on the right) and right click the background layer, add alpha channel. They you can select everything and hit delete. The chequered squares means that that part of the image is transparent