I'm not very good at photoshop but I wanted to know how do you unite two shapes (circles) that have different colors? When I unite my two shapes the second shape takes on the color of the first circle. Both shapes are separate layers made using the eclipse tool. Any help would be greatly appreciated.
When you unify more than one shape, you create one shape. Therefore you can't have multiple colors applied to a single shape (other than gradients). If you need two colors, you need 2 shapes.
You could achieve what you are after with 2 shapes, a layer group and a mask on the group.
Option 1 (Meh)
To unite 2 circles of different colors, you can simply add a layer mask on each of your circle and trace an oblique mask that goes right in the middle of them.
Start with one, then duplicate that layer mask on the second circle for precision; to do this, click the layer mask you did while holding option+command, and drag that mask to the other layer where the second circle is.
Then click that new mask and go in the menu "adjustment", then select "invert".
Then if you want to unite these 2 circles of different colors, you can add them to a layer group... and right-click it, and merge! To add a group, select your 2 layers, and click the folder icon on the bottom of the layers window... OR click the folder icon and drag the 2 layers into it.
If you want to keep the option of changing the colors later, you don't need to merge that group!
Option 2 (Very bad)
Unite the 2 circles together, then do an oblique selection as shown above.
Then go in "adjustment", then "Image" then select "hue and saturation". Adjust your color.
This is a very bad technique. The one above leaves you way more room to adjust your colors and change them later.
Option 3 (Best option)
You could also do this easily in Illustrator using the Pathfinder tool, and then place that new shape in Photoshop. You'll still have 2 shapes though but you'll get the illusion it's one single shape with 2 colors. At least they won't be rasterized if you do it this way.
If you used more complex shapes, you can export your paths in ai and modify them in Illustrator if it's easier, and then place them back. There's way more tools and control to work with paths in Adobe Illustrators. If you don't have it, I guess you can try InkScape, it's free.
You could always do this in Photoshop too but you'll need to work on your paths once they're merged, and remove/add new paths to split them. Then you won't have 1 single shape... but you'll get 2 circles with different colors the way you want.