It depends on the show.

In some cases, shows are written to be anthologies of stories. The characters stay similar across episodes and seasons, but the isn’t really an overarching plot. Sitcoms are known to use this a lot. Plot across episodes is mainly done to give writers something new to write.

In other cases, several plot lines are happening at once which resolve at different times. That way, there is always a plot having something happening even if other plots end or hit a resting point. A lot of soap operas did this.

Finally, there can be one overarching plot that gets resolved, but then another plot starts to take its place or the show ends. A lot of modern science fiction is written that way.