Madeira is one big main island and a few small ones, about 1,000 km off the coast of Portugal - that's like going from London to Marseille. The whole island is the top of a giant volcano that started underwater. The land is very steep and green because lots of rain falls on the high middles of the island.
The Azores are even further out - 1,400 km from Lisbon. They are nine separate islands, scattered across the ocean, and each one has its own volcanoes. Some islands have big crater lakes where you can swim in warm water heated by the earth. On Pico Island, there is a volcano taller than any mountain on mainland Portugal.
Because the islands stick up out of the deep ocean, they are surrounded by sea creatures that don't live nearer the shore. Bottlenose dolphins, sperm whales and giant manta rays swim past every year. From the cliffs, you can spot whales blowing water into the sky.
People on the islands have their own way of life. Madeira is famous for a cake (a kind of soft, honey-and-spice sponge) and for flower festivals in spring. The Azores are famous for cheese, dairy cows, and the gentle, slow pace of island life. Both places feel more like a sea adventure than a normal European holiday.