The Fanatic badge, for example, requires consecutive days. If you log in exactly every 24 hours it will work. Your profile displays consecutive days as well as total days.
Specifically, though, StackExchange's "day" is the UTC (GMT) day. It's easiest if you think of the week as being divided into 7 24-hour chunks. Log in once during each chunk and you're good for the week.
For example: If you visit at 6:00:00 UTC one day, you have to wait at least 18 hours and at most 42 hours (41:59:59.999..., until 23:59:59.999... the next day) get the second consecutive day. If you visit right at that last second then you have to visit again within exactly 24 hours to get the third consecutive day. And so on :)