Successive means one after another with nothing in between — an unbroken run. "Three successive wins" means win, win, win, no loss in the middle. It almost always sits before a plural noun and almost always has a number or *several* in front of it: "four successive days", "two successive governments", "several successive attempts".

The unbroken part is the whole point, and it is why the word is so common in sport, weather reports, and business news, where a run of the same result is the story. "Prices rose for a fifth successive month" tells you it happened in month one, two, three, four and five — not five times scattered across the year.

Be careful with a second, quieter use: "successive versions of the software" means each one came after the last, and usually that each one built on the last. Here there is no counting; the feeling is a chain of improvements. Both uses are slightly formal but not difficult — you will meet them constantly in newspapers. In speech people more often say "three days in a row" or "three days straight", which is exactly the same idea in easier words.