The first-Thursday rule is the ISO 8601 rule that defines week 1 of any year as the week containing the first Thursday of that year. Equivalently, week 1 is the week containing 4 January.
Why Thursday
Thursday is the 4th day of the ISO week, which runs Monday to Sunday. Anchoring week 1 to the first Thursday guarantees that at least 4 of the week's 7 days fall in the new calendar year. This makes the majority-of-days assignment unambiguous.
The 4 January equivalence holds because 4 January is always in the same ISO week as the year's first Thursday. Both definitions produce the same result.
Contrast with the US convention
US week numbering assigns week 1 to the week containing 1 January, regardless of weekday. Under that rule, week 1 may contain as few as 1 day of the new year. The 2 conventions disagree for roughly half of all years, producing different week-1 dates for the same calendar.