"NAND" gates are inverted AND logic gates. NAND Flash is the physical component on the SSD responsible for storing your data. NAND Flash memory can be seen as sort of an opposite of RAM (Random Access Memory), which—as you all know—is a type of fast, volatile memory used for storing data temporarily. Non-volatile memory is represented in "permanent" storage -- hard drives, solid-state drives, and Flash memory (USB, SD) devices. The term "NAND" is often used as a truncated term to refer to the dies on the SSD (for example, Micron's 128Gb NAND Flash).
Flash memory was originally based on EEPROM, or Electrically Erasable Programmable Read-Only Memory, and has been adapted for use in modern SSDs. If you're familiar with the term "P/E cycles" -- or Program/Erase cycles -- as it pertains to SSDs, you'll already have a basic understanding of what "electrically-erasable / programmable" means.
There are various types of NAND Flash and it is constantly evolving. We'd recommend reading this page of our SSD Architecture post to learn more. The basic NAND Flash types are SLC, MLC, and TLC, each defined individually in this dictionary.