This is a high-level overview of the cryptography used in Polkadot. It assumes that you have some knowledge about cryptographic primitives that are used in blockchains generally such as hashes, elliptic curve cryptography (ECC), and public-private keypairs.
For detailed descriptions on the cryptography used please see the more advanced research wiki.
Keypairs and Signing
Polkadot uses Schnorrkel/Ristretto x25519 ("sr25519") as its key derivation and signing algorithm.
Sr25519 is based on the same underlying Curve25519 as its EdDSA counterpart, Ed25519. However, it uses Schnorr signatures instead of the EdDSA scheme. Schnorr signatures bring some noticeable benefits over the ECDSA/EdDSA schemes. For one, it is more efficient and still retains the same feature set and security assumptions. Additionally, it allows for native multisignature through signature aggregation.
The names Schnorrkel and Ristretto come from the two Rust libraries that implement this scheme, the Schnorrkel library for Schnorr signatures and the Ristretto library that makes it possible to use cofactor-8 curves like Curve25519.