Integrated circuits are prevalent in so many forms across electronics, it's hard to cover everything. Here are a few of the more common ICs you might encounter in educational electronics.
Logic Gates, Timers, Shift Registers, Etc.
Logic gates, the building blocks of much more ICs themselves, can be packaged into their own integrated circuit. Some logic gate ICs might contain a handful of gates in one package, like this quad-input AND gate:
Logic gates can be connected inside an IC to create timers, counters, latches, shift registers, and other basic logic circuitry. Most of these simple circuits can be found in DIP packages, as well as SOIC and SSOP.
Microcontrollers, Microprocessors, FPGAs, Etc.
Microcontrollers, microprocessors, and FPGAs, all packing thousands, millions, even billions of transistors into a tiny chip, are all integrated circuits. These components exist in a wide range in functionality, complexity, and size; from an 8-bit microcontroller like the ATmega328 in an Arduino, to a complex 64-bit, multi-core microprocessor organizing activity in your computer.
These components are usually the largest IC in a circuit. Simple microcontrollers can be found in packages ranging from DIP to QFN/QFP, with pin counts lying somewhere between eight and a hundred. As these components grow in complexity, the package gets equally complex. FPGAs and complex microprocessors can have upwards of a thousand pins and are only available in advanced packages like QFN, LGA, or BGA.
These ICs are usually smaller than the microcontrollers, or other ICs on a circuit board, with pin counts in the three to twenty range. DIP sensor ICs are becoming a rarity, as modern components are usually found in QFP, QFN, even BGA packages.