The way television scheduling work in Japan actually makes a lot more sense than how it's done in the United States.
Ask yourself, in the US, how many TV seasons do we have in a year? We sort of have a very long main season, where TV series will often have up to 24 episodes scattered haphazardly over a roughly 36-week period (without well-defined boundaries) with random hiatuses so you often don't know if they'll be a new episode or not unless you make an effort to check. There's also a short "summer season" that nobody really cares about. Midseason cancellations and replacements are common; many series will be cancelled after one or two episodes.
With anime, the year is divided into four periods called "cours". They're super simple to remember: winter is January - March, spring is April - June, summer is July - September, and autumn is October - December.
Most anime series are classified as one-cour or two-cour. There are higher-cour series, but not many.
A one-cour series runs continuously for up to 13 weeks (starting in early January, April, July, or October). Some are shorter, most commonly 11 or 12 episodes, starting a week late or finishing a week early to compensate. Breaks during the run are very rare: they almost always run completely continuously. They also almost always keep the same OP (opening) and and ED (ending) songs and sequences for the entire run.
Two-cour anime can be classified as Winter/Spring, Spring/Summer, Summer/Autumn, or Autumn/Winter. They run up to 26 weeks, though 23-25 is more common. The often have a 1 or 2 week break between cours, especially Autumn/Winter because pretty much everything shuts down for New Years. The first cour will often end with a cliffhanger or other significant episode.
Two-cour anime frequently mark the transition between cours with a change to the OP and/or ED. In a way, think of it as a new season, even though there's at most a 1 or 2 week gap between episodes, or often no gap at all. These changes help build excitement and try to get people to tune back in for the second cour despite being tempted by all the new series starting at a the same time.
There are exceptions like 3-cour, 4-cour, and a few series that run continuously for years with never a break of more than a couple weeks. But one-cour and two-cour are by far the most common and most important.
Midseason cancellations are also very rare. Usually the creators know exactly how many episodes they have to work with, and will aim to tell a complete story within that span of episodes, sometimes dropping hooks for a potential sequel. If a series gets a sequel, it may be years later, and will usually have a modified title.
tl;dr - the three-mouth cour system is a very important concept in anime & OP/ED changes often mark the change between cours