The seasons are driven by complex interactions, and while most answers already given here include a piece of the puzzle, none paint a complete picture, so I'll try to integrate some of the major factors.
The seasons on any planet (not just Earth) are governed primarily by the following characteristics:
- The tilt of the planet's rotational axis relative to its orbital plane (obliquity of the ecliptic)
This is the main driver of seasons on Earth and affects both the duration of sunlight received in a day (the hemisphere more exposed to the sun will receive more hours of sunlight) and the directness of that sunlight (which affects absorption and reflection). The degree of tilt affects the relative strength of the seasons; for instance, if Earth's obliquity of the ecliptic was at or above 45 degrees, then when one hemisphere was tilted toward the sun it would experience dramatically longer, hotter days, while the opposite hemisphere would endure equally long nights.
This plays almost no role in Earth's climate as our axis of rotation is stabilized by our Moon. If we lacked that stabilizing feature, Earth's axis would slowly wobble due to a variety of factors (sloshing of the ocean, flex in the crust, etc.) such that the tilt at one point in the orbit would not be the same at another point in the orbit. This could lead to significant variations in the length of seasons which may be difficult to predict.
- The eccentricity of the planet's orbit around its star (variation in distance from the star)
This directly affects how much solar energy is available to be absorbed by the planet. Earth has a nearly circular orbit, so we see less than 7% variation in available sunlight between perihelion and aphelion and this mechanism is therefore not a significant factor in our climate. A more elliptical orbit would result in a much warmer average temperature when close to the parent star, as well as an increased orbital speed at that closer distance. Conversely, the average temperature would be much lower and the orbital speed slower at the greatest distance from the star.
This doesn't mean you'd have hotter, shorter summers and longer, colder winters, because summer and winter are mainly defined by axial tilt. A more eccentric orbit would affect the relative strength of the seasons. If the eccentric orbit was somehow coupled with variation in the planet's axial tilt, (e.g. the axial tilt wobbles once every 2 orbits, such that at perihelion on year 1 it might be summer in the north, and at the same point in the orbit on year 2 it's winter), then you might see more dramatic variations season length.
- Variation in the star's luminous output (solar cycles, star type)
Stars are not static objects, and their luminous and spectral output varies depending on their size, type, age, and other factors our scientists have yet to fully understand. Our own Sun operates on a roughly 11-year cycle of sunspot activity which has been shown to influence the seasons. A more wild star, or a multi-star system, would have an even greater impact on the intensity and possibly the length of seasons.
- Variation in the absorptivity, heat capacity, emissivity, and reflectivity of the planet (ability to capture, store, emit, and reflect heat from the parent star)
Earth is roughly 75% water, which absorbs sunlight readily and stores heat easily. Our clouds reflect light readily, but can also prevent heat from escaping back into space. As clouds move across the planet's surface, and as the planet rotates on its axis, different materials are exposed to the sun, the local reflectivity and absorptivity changes, wind and evaporation come into play, and we get the complex interactions that we call weather. This variation in the planet's ability to absorb and release heat is possibly the second-greatest driver of the seasons on Earth, and it is so staggeringly complex that many people still don't agree on how it works (climate change, anyone?).
Venus is locked in an endless, brutal summer, not because it is closer to the Sun, but because its own weather readily absorbs heat but does not release it easily. Mars, by contrast, is entombed in eternal winter largely because it has almost no atmosphere to speak of, and thus can't hold onto what little sunlight it receives.
Mixing and matching the above factors could yield a variety of unusual climates and seasons, without the need for bizarre celestial circumstances or magic. Although, it is fair to say that using the weather to justify unusual weather may require a bit of handwavium since that subject is so complex.