What is Arbitrum?
Arbitrum is a Layer 2 scaling solution for Ethereum. You can use the Arbitrum chain to do anything you can do on Ethereum - use Web3 applications, deploy smart contracts, etc., but the transactions will be cheaper and faster. Arbitrum Rollup inherits Ethereum-level security and solves the problem of Ethereum's low TPS.
How does Arbitrum rollup work?
The basic idea of Arbitrum rollup is this: the Arbitrum Rollup chain operates as a sub-module in Ethereum. Unlike regular Layer 1 Ethereum transactions, we don't need an Ethereum node to process every Arbitrum transaction; instead, Ethereum takes a 'not guilty until proven guilty' approach to Arbitrum. Layer 1 initially 'optimistically assumes' that activity on Arbitrum follows proper rules. In the event of a breach (i.e. someone claiming 'now I have all your money'), a dispute can be filed on L1; fraud will be proven, invalid claims will be ignored, and malicious parties will be financially penalized. This ability to adjudicate and prove fraud at L1 is a key, fundamental feature of Arbitrum, and how and why the system inherits the security of Ethereum. So we can use Ethereum to prove fraud on Arbitrum.
So how can Arbitrum prove that if there is fraud? This is where the 'rollup' part comes in. The data entered into the Arbitrum Rollup chain (i.e. users' transaction data) is published directly on Ethereum. So, as long as Ethereum itself is running securely, anyone interested can see what's happening in Arbitrum and have the ability to detect and prove fraud.
The parties that move the state of the Arbitrum chain forward on L1 (i.e. make claims about the state of the chain, dispute the claims of others, etc.) are called validators. Becoming an Arbitrum validator requires no special permission, only requires users to run the open source validator software (and stake ETH when/if they need to take action). Furthermore, the chain remains secure as long as there is only one honest validator; that is, it only takes one non-malicious fraudulent prover to catch any number of malicious troublemakers. Together, these properties make the system 'trustless'; users do not rely on any particular designated party to keep their funds safe.
The only delay Arbitrum users experience with their transactions is the 'withdrawing' process - transferring their funds from Arbitrum back to Ethereum; if users withdraw directly from Arbitrum to Ethereum, they typically have to wait 1 week for their funds to be received on L1. However, users can bypass this delay time entirely (possibly for a small fee) if they use the ExpressBridge app. Anything else a user does — for example, depositing funds from Ethereum into Arbitrum, or using a dapp deployed on the Arbitrum chain — will not result in this delay period.
The experience of using Arbitrum is very similar to that of Ethereum. Different Layer 2 protocols emphasize and optimize for different things; Arbitrum was created with Ethereum compatibility as a top priority. This means that users can use Arbitrum in all their favorite Ethereum wallets; developers can use all their favorite Ethereum libraries and tools to build and deploy contracts; in fact, most of the time, the experience of using Arbitrum is the same as using Ethereum. with the same experience (with the important exception that it's cheaper and faster).
A lot of development work has gone into achieving this level of Ethereum compatibility. But at its core: Arbitrum itself uses a fork of Geth and modifies it to convert it to a trustless Layer 2. This means that much of the code running in Arbitrum is the same code that runs in Ethereum. This cutting-edge approach is called Nitro.
Arbitrum AnyTrust Chain
Arbitrum Rollups are primarily designed to avoid introducing any centralization or trust assumptions, so this is a clear, strict net win for the Ethereum ecosystem. However, decentralization has a (literal) price, and not all applications and users necessarily want or need to pay that price. For dapp use cases with different security considerations, different tools in the Arbitrum suite are appropriate; namely, the Arbitrum AnyTrust chain!
Arbitrum AnyTrust chains do not have the same decentralized/trustless/permissionless security guarantees as Rollup chains and thus can offer lower fees. Rollup and AnyTrust are similar in many ways, but with one key difference: in Rollup, all data is published on L1 (allowing anyone to join permissionlessly as a validator), while in AnyTrust, data is on-chain under management. In the event of a challenge, the AnyTrust chain reverts to 'rollup mode'; the safe assumption here is that at least 2 committee members are honest (i.e., they will provide data when necessary). In the happy/common case, keeping the data offline means the system can charge the user less. For fully decentralized applications that require high transaction throughput and do not require rollup provisioning, AnyTrust may be a sensible compromise.
So there is more than one Arbitrum chain on Ethereum. The fact that multiple chains can run in parallel is an important advantage of off-chain scaling techniques. Currently, on the Ethereum mainnet, there are 2 Arbitrum chains: an Arbitrum Rollup chain called 'Arbitrum One', and an AnyTrust chain called 'Nova'; users and developers can choose what suits their security/transaction cost needs anything.