Learn Swedish: Den andra formen visar en annan detalj. - The other shape shows another detail.

It comes down to whether the phrase is definite (the) or indefinite (a/an). In den andra formen (the other shape), the phrase is definite, so we must use the definite form andra. In en annan detalj (another detail), the phrase is indefinite. Since detalj is an en-word, we use the indefinite en-word form annan.

Why is it den andra formen instead of just den andra form?

This is the classic Swedish double definite rule. When you have a freestanding definite article (den) and an adjective or determiner (like andra), the noun itself must also take its definite ending (-en). You are literally saying "the other shape-the".

If andra is used for the singular "the other" here, how would I say "other shapes" in the plural?

You still use andra! The word andra serves double duty: it is the definite singular form (as seen in this sentence) AND it is the plural form for both definite and indefinite nouns. So "other shapes" would simply be andra former.

Since detalj is an en-word, what would the phrase look like if we used an ett-word instead, like exempel (example)?

If you wanted to say "another example", you would need the ett-word form of annan, which is annat. The phrase would then become ett annat exempel.