Where is the genetic material in a eukaryote stored?
The most obvious answer, and the one that it usually being sought when this question is asked, is that genetic material is stored in the nucleus of the eukaryotic cell.
The name "eukaryote" means "good kernel" - this is a quaint but outdated description of how eukaryotes looked when people first saw them in a microscope. The role and structure of the nucleus was unknown to them; they just saw what looked like a small kernel, or grain, of rice or wheat. The nucleus does vaguely resemble this shape, but the name doesn't do much good in describing its role or function.
Most of the DNA is stored in the nucleus because the nucleus is a highly-regulated and therefore well-protected place that minimizes the amount of "inappropriate" contact, helping to keep the DNA intact and free from outside influences that would damage or alter it. It's not a perfect setup, but it's more regulated than a prokaryote is.
DNA is also found in other parts of the eukaryote, such as the cytoplasm, and especially in the mitochondria and chloroplasts. The presence of DNA in these organelles suggests that they were once separate living organisms, which later "teamed up" with a eukaryotic cell in a process called endosymbiosis - literally "living together on the inside." The mitochondria and chloroplasts found it more advantageous to live inside the cell, providing it with nutrients in exchange for protection.