I am having some difficulties trying to delete the space where I created a simple '+' icon. Here are some images for reference.
I hope this explains what I am trying to accomplish. Sorry for not using technical terms, I do not know them!
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You don't mention the software package. The generic term(s) for using one shape to alter another one in this manner are: "intersection" "subtraction," "union/merge," and "XOR (eXclusive OR)".
You can search the documentation for your software for terms that look like these. Often they are under an "object" menu and in adobe products, they are on a palette which may be accessed via the "window" menu ("Pathfinder").
Select both shapes and click the Minus Front button on the Pathfinder Panel.
But Horatio is correct, looking in the help files for "subtract shape" will result in steps to do what you want. Help Results
In Photoshop you have to select the shape you wanted to delete some of the area. Make sure the Direct Selection Tool is active. Then create the shape you wanted to intersect on the shape, then select subtract from path operations as for your question.
There are other options like: Combine, Subtract, Intersect and Exclude