This is an intellectual movement in Europe with which is associated names like Claude Levi-Strauss, Ferdinand de Saussure, Roland Barthes, and Jacques Lacan.
For having a better understanding of structure and structuralism, it would be better to learn what a Concept is. A concept is constituted of five essential elements : name or title, examples, attributes, degree of attributes and finally definition or generalization. Examples may be positive or negative, attributes may be essential or non-essential. All other terms are self explanatory. A structure is set of concepts.
This is a preliminary discussion about how we understand events, or phenomena, by looking at their relations with other events, or other phenomena. Unless we find out deeper or less visible 'structures' we do not get at the full meaning of events, or any sequence of events.
Just consider this: a critic has pointed out that suppose there is a train that travels between station A and station B, at 8.15 a.m. every day. The rake (the combination of coaches) is not the same everyday. Neither is the engine. But it is always known as the "8.15 Express". If it is late and leaves at 8.22 a.m., it is still the 8.15 Express, running late. If there is a mishap on the tracks in between A and B, and the passengers are transferred to another train, they will still have travelled by the 8.15 Express. If the railway lines have been broken somewhere on the way, and the passengers are put on to a bus from the point where the train was stopped, and taken to B by the bus, they will still have travelled by the '8.15 Express'. In other words, the '8.15 Express' is a train scheduled to hold a particular place on the railway timetable because its meaning comes from its relation to other trains with their own properties.
This merely begins our approach to 'Structuralism'. The example above just helps us to see that here the meaning of the 8.15 Express is closely linked with the railway timetable, other trains (between stations X and Y) and does not really have anything to do with the physical coach, the engine, the engine-driver who will be different on different days, and so on. The 'structure', of the railways, etc., gives the 'meaning'.
You can see from this brief discussion that "train" or "8.15 Express" does not mean anything related to the object. If a child asks, "What is the 8.15 Express?" , how will I explain it? ("A train" - "What is a train?" - "That lo-o-ng thing - can you see there?", and so on.) For us we can understand that there is a public transport system which 'creates' trains by arranging the whole system. That whole system is the basic structure.
Applying this to other aspects, -- as did the anthropologist, Claude Levi-Strauss, to social relations-- we can unearth the very many underlying 'structures' that regulate our social lives.
In our little discussion we saw that the name '8.15 Express' means a lot. That is, it means that it is not the "10.30 Passenger Train",etc. In other words, the name, or what Saussure called the 'signifier' , has its own properties.