“Initial claims” is just that — the first time you go to the unemployment office after you’ve been laid off.
(One angle that I don’t see discussed even here on FR is the increasing tendency of employers to nudge you into quitting (via increasing workload, setting unrealistic goals, death by a thousand cuts, etc.) or inventing reasons to fire you (via impossible to disprove “conflicts with other workers”, e.g.) rather than lay you off, as that more often than not disqualifies you from getting unemployment insurance.)
I think “jobs created” is something they get via the weekly BLS survey of some employers. (The question of “jobs created or saved” via Porkulus or the Census is a completely different matter.) It might be as simple as comparing how many people are employed this week to how many were employed last week.
One way think about the two numbers is to keep in mind that a new business is going to add to Jobs Created, and an existing business that lays off people is going to add to Initial Claims, and there’s no paradox there. Also, big businesses add new jobs one hand (for a new product, for example, or a new office somewhere) while at the very same time they are laying off employees, closing an office, etc.
Thanks!