"Dirty" is used to refer to inbred lions, and "clean" lions have no inbreeding at any point in their entire heritage. You can view this by clicking on "View Full Heritage" on a lion's page. If it is inbred, there will be a notice. You may need to click through the family tree to explore the entire thing. Here is one of my lions who is dirty:
https://www.lioden.com/heritage.php?type=1&id=794160148470
And here is one who is clean:
https://www.lioden.com/lion.php?id=794184103931

Note that clean vs dirty has NO effect whatsoever on gameplay. Some people just have a personal preference to have clean lions.

edit - people use "potato" to refer to boring looking brown lions with no interesting markings.

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Inbreeding as in incest. This happens if lions who are closely related breed together. For example, the first lion I linked above has the same lion "Snow cloud" as both his father and grandfather. So he is inbred.

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What’s inbreeding?

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When a lion breeds with a lions who are closely related.
Eg. A father lion having cubs with a daughter lion or grandmother or granddaughter.

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