If our solar system was like 90% of most we would have two or more stars in it. In that case, we would have to know the mass of each star and the distance between them in order to compute the path of a planet. we also would have to know the angle between the spin axis of the planet and the plane defined by the two stars to say something about possible climates and seasons.
So from knowledge simply that a star system is a binary one, one could not really say anything about seasonality on the planet. More info is needed.

Neat question!I'll help you find the answer. First, find out what causes us to have seasons. Once you've found that out, think about how the sun is involved in our seasons. Is it only the movement of the sun that causes seasons? Is it only the movement of the Earth? Once you've figured all that out, draw a picture of the positions of the earth and the sun during the different seasons. Then, draw another sun in different places in the solar system and figure out how it would affect the seasons on earth. I'd love to hear your answer. Let me know how it goes!

You could actually answer this question yourself by building a simple model. First, take two large balls like basketballs or soccer balls and place them on the floor near each other. These will represent your two suns. Next take a smaller ball, like a tennis ball, and place this on the floor away from the other "suns". This will be your planet. Next, draw an elliptical orbit in chalk on the floor (or scratch it on the ground if you do this outside). You should decide if your orbit will pass between the two suns or go completely around them. You could actually draw both to represent the two scenarios; one where the planet passes between the suns and the other where it goes around them both.

Now place the "planet" in different places around each orbit and answer these questions.

If your planet is between the suns, will the days be longer? If there were only one sun, would there be a difference? Will the days be hotter or colder? What season does this represent, summer or winter? When the planet is on the far end of the ellipse orbit, what season is this? Will the days be much longer or shorter? Will the difference be very very large compared with summer when the planet is between the suns?

For advanced students: An ellipse has two foci. The model above assumes that both suns or at least one is located in one of these foci. Place one sun at each foci and answer the above questions again.

A word on models: What you have done is what scientist do all of the time; build models to understand things better. The model you just built is both a physical model and a conceptual model. It is physical because it is actually there unlike a computer model or a mathematical model. It is a conceptual model because you do not do any measurements even though you could.

It would depend on where the suns were located in relation to each other and to the other planets in 0ur solar system. Having two suns would also change the gravitational forces on our planets. For instance, if the force on the earth is taken as 1, the comparative force for the sun is 27.9. That means that if, for example, a person on earth weighed 100 pounds, then the weight of that person on the sun would be 100 x 27.9 = 2790 pounds. The only other gravitational force exerted in our solar system that even comes close to this is Jupiter, which would exert a gravitational force of 2.64 (going along with the example). So, if there were another sun in our solar system, it could potentially change the planets orbits, or even change the seasons on the planets. In terms of whether it would be summer year-round or if the sun would ever set, I think this would depend largely on where the other sun was located in relation to earth and our current sun.

Hope that helps!

If the Sun were part of a binary star system then things would be definitely be different. Each star would rise and set as the earth rotates about its axis. Depending on how things are arranged there could be times when both stars can be seen, times when only one star is "up," and maybe times when neither star is "up." How an extra star would effect the weather and climate would also depend very much on how bright the extra star is and how far away it is. There are many, many possibilities.