Ethereum 1.0 can only process 7-15 transactions per second, the goal of sharding is to partition all network computational resources into shards, so that a node (a single computer as a peer connected to the network) doesn't have to process (download, compute, store, read) every transaction in the history of the blockchain, in order to make a new transaction (write and upload) or otherwise participate in securing and using Ethereum; rather a node can just participate in a single shard, or more if it so chooses. Multiple shards are handled separately by different subsets of securing participants, aka securitors (which include notaries, proposers, miners and validators). The primary goal is a massive scalability improvement, potentially exponential in phase 6 of the roadmap; and probably at least a 100 fold increase in transactions per second with earlier phases. Quadratic sharding involves having shards at a depth of at most 1 from the main chain, that is, there are no shards within a shard, or a manager shard managing sub-shards; whereas exponential sharding has shards within shards, recursively.
Each one of the shards (likely 100 live in phase 1) will have as high a capacity (and likely more with phase 1) than the current existing Ethereum chain. A smart contract will exist on the main chain, called the sharding manager contract, which manages how data and transactions in shards are accepted as valid by the main chain, via notaries voting on the validity and data availability of collations (collections of data and transactions, analogous to blocks, but where they occur more frequently than blocks) in shards, and proposers proposing blobs (analogous to transactions but without execution in phase 1) which are collected into collations.
For information on sharding, refer to (sorted roughly from the most recent/important information to less recent):
Implementations under development include:
Enshrined-in-consensus: see https://ethresear.ch/t/sharding-phase-1-spec-retired/1407/18.