Organelles are specialized membrane-enclosed structures in the cytoplasm of eukaryotic cells. The nucleus, which contains DNA in the form of chromosomes, is the most characteristic eukaryotic organelle.The nuclear envelope is connected to a system of membranes in the; cytoplasm called the endoplasmic reticulum (ER).he nucleus, which contains DNA in the form of chromosomes, is the most characteristic eukaryotic organelle. The nuclear envelope is connected to a system of membranes in the; cytoplasm called the endoplasmic reticulum (ER).