Let me start with an analogy:

On mainnet, when Alice wants to send to Bob, she has 2 options:

1. Sign transaction and broadcast it (Bob will find it on mempool)
2. Sign transaction and send it to Bob (Bob can broadcast it immediately or later)

Let’s call it Send and Send Async respectively.

In Ark there also 2 options for Alice (assuming Alice already has a VTXO).

Send:

• Alice asks the ASP to pay Bob
• ASP waits for other payment intentions
• ASP prepares a new round tx with a new VTXO belonging to Bob and shows it to Alice
• Alice signs a forfeit tx using a round tx output as input (making it atomic)
• Alice needs to be online during the round
• ASP liquidity is required immediately

Send Async (or Out-Of-Round):

• Alice prepares a redeem tx (spends Alice’s VTXO into a Bob’s VTXO)
• Alice ask the ASP to co-sign the redeem tx
• Alice sends the redeem tx to Bob (it doesn’t matter how)
• Bob can use this redeem tx to join a round now or later (up until Alice’s VTXO validity)
• Alice doesn’t need to be online during the round
• ASP liquidity is required later (only when Bob joins the round)
• Up until joining the round, Bob is trusting Alice + ASP don’t collude to double spend him

For a more detailed explanation, check Payments and Out-of-Round Payments on arkdev.info

Related: Ark explainer