API
An API is a defined interface that lets one software system request data or actions from another, using a documented set of operations rather than a human interface.
Why it matters to a non-technical decision
Whether two systems can be connected, and how much that connection costs, is mostly determined by what their APIs allow. A tool with a well-documented API that exposes the records you need is straightforward to integrate. A tool without one has to be worked around, and the workaround is usually where the fragility and the ongoing maintenance cost live.
What to ask a vendor about it
Three questions cover most cases: which records and actions the API exposes, what the rate limits are, and whether your plan includes API access at all. The third one is a common surprise, because API access is frequently gated behind a higher tier than the one a business already pays for.
Direction of the request
An API call is initiated by the system that wants something: it asks, and waits for an answer. That is the opposite of a webhook, where the system holding the information announces a change as it happens. Most connected processes use both.