Do your digital photos look blurred and grainy?

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The other day I downloaded a cute cat picture for my daughter. She saw it on MyLot and wanted to look at it closer. So, I obliged. It looked great online. But, when I got it on my computer, turned out to be a tiny pic. So I blew it up and it warped and distorted and became blurred. Does that happen to your digital pics when you try to edit or adjust the size? How do you keep that from happening? Does that happen to pics taken by your digital camera? Would love to know a solution. Thanks!