This is a huge topic because there are so many kinds of arthropods. There are three broad groups to consider; insects, crustaceans, and the chelicerates (spiders and their cousins). Spiders and insects are terrestrial animals and so have similar to solutions to the problems associated with terrestrial ecosystems. Crustaceans are the 'insects of the sea' and their organ sysems are adapted to aquatic ecosystems.
All three groups show similarities in the types of organ systems that they have even if they have obviously evolved them separately. Complex animals need a musculoskeletal system in order to move, sense organs and a nervous system to find food and escape predators. They need a respiratory system to get oxygen to cells and get rid of waste gases and a digestive system to process food and get rid of wastes. We have eyes, ears, mouths, and senses of touch and taste plus a nervous system to coordinate all the information and send and receive messages to the body. and arthropods also have these, though they evolved separately. The differences between insect and crustacean senses and nervous systems involve different solutions for terrestrial and aquatic environments, just like the differences between mammals and fish.
The biggest difference between arthropods and other groups including our own is that they opted for an exoskeleton for protection and that included jointed coverings over the limbs. The name arthropod means jointed leg and indicates a basic structural form that determined the set up of organ systems in both insects and crustaceans. Muscles have to be hung on the inside of the skeleton and this limits the size of the organism. Most importantly, the exoskeleton has to be shed in order for the animal to grow. Although the arthropod phylum is amazingly diverse and successful, the biggest animals that have ever lived have had an endoskeleton like us.
Another important difference between arthropods and vertebrates is the placement of the digestive and nervous systems. We have a dorsal nerve chord and a ventrally located digestive system. In arthropods, the stomach and intestine run down the back of the animal and the nerve chord likes along the ventral surface. In other ways the digestive systems are similar: start with a mouth at the head end, armed with chewing mouthparts (mandibles in insects and crustacean and chelicerae in the spiders). Food is swallowed and passed down an esophageal tube to a stomach where digestion occurs and then on to an intestine where further digestion plus absorption occurs and then on to a lower intestine and an anus where waste materials are excreted. Forms may vary but because the function is basically the same, digestive systems are roughly comparable throughout the arthropods.
The system that varies the greatest between aquatic and terrestrial lifestyles is the respiratory system. Crustaceans have to extract dissolved oxygen from water while insects have to extract oxygen from the air. Crustaceans have gill-like structures to allow their circulatory fluids to pass close to water and allow oxygen to diffuse into the tissues. Many crustaceans have a heart and blood which moves both gases and nutrients to all their body cells and remove wastes. Some have the equivalent of veings and arteries but some merely have an ' open' system where the fluids slosh around the body in cavities. On land insects use openings called spiracles to take in air into tracheal systems that carry the air close to circulatory fluids. Insects also use a circulatory system to carry nutrients to cells and waste products away.
So even though arthropods are built on different body plans, they must still carry out all the functions necessary for any animal to survive: movement, growth, reproduction, food gathering, digestion, excretion, respiration and circulation. For each of these functions there are organs, organised in organ systems, to carry out these functions. In the end it is all about survival. Arthropods have found a multitude of ways to survive in almost every conceivable niche in almost all terrestrial and aquatic habitats. But underneath the functions are the same so similar organ systems have evolved.
Information sources: Meglitsch, Invertebrate Zoology and Borror & DeLong, Introduction to the study of Insects.