No, it's not. You can use the free plan as long as you want without credit cards. Just be aware that if you reach the free plan limits in a site and your Account is not associated with a credit card, your editors won't be able to add new content to their website until you do it.
A record represents a single piece of information you store within a site. Think of it like a database-like entry. It can be anything: a blog post, a category, an image gallery, a staff member in your about page. Within each site, you can configure the specific fields required for each kind of record.
Every website you manage with DatoCMS needs a separate site. Each site is completely isolated from the others, with different data and editors, and is accessible from a different domain. You can manage your sites from your Account dashboard.