An organ is a fully differentiated structural and functional unit in a multi-cellular organism (unicellular organisms have just one cell and hence can not possess tissues or organs), composed of one or a group of tissues, and has one or more particular functions. For example, a kidney is a fully functional organ and filters the blood and also generate hormones that regulate the blood production in body. Some other examples of organs are heart, eyes, kidney, etc.
Tissues are made up of cells (identical or similar type of cells) and each tissue carries out a specific function. Some examples of tissues are, nervous tissues, muscle tissues, epithelial tissues, etc. Tissues are intermediate organizational level between cells and organs. In other words, similar cells make up tissues, similar tissues make up organs.
Therefore, it can be stated that an organ is always made of tissues, but tissue is not always considered an organ.
It is the same as saying chocolate cake is always made of chocolate but chocolate in not always made into a chocolate cake :)
What is meant by the following statement: An organ is always made of tissue, but tissue is not always considered an organ.
It has to do with the level of organization in living things. The levels of organization in living things are cell, tissue, organ, system and organism.
Cells are the basic unit. Groups of similar cells join together to make tissues. For example, muscles are tissues, made up of muscle cells joined into muscle fibers.
Groups of different tissues make up organs. For example, the heart is an organ, it's made up of muscle tissue, nervous tissue and connective tissue.
Organs are made up of tissues. Tissue by itself does not make up an organ.