Salesforce. A new Calendly booking creates or updates a lead or contact and logs the meeting as an activity against it, using whatever default field mapping the integration ships with.
HubSpot. Works the same way: a booking creates or updates a contact and logs the meeting, and Calendly scheduling links can be embedded directly into HubSpot workflows and emails.
Slack. Posts a notification into a channel you choose when someone books, reschedules, or cancels, mainly useful for visibility rather than record-keeping.
Everything else, usually via Zapier. ActiveCampaign and most other marketing or CRM tools aren't native Calendly integrations; getting a booking into them typically means a Zapier or Make step in between.