Prefixes are stuck to the front of words, suffixes are in the back. You're asking about the suffix. :)
The base words here are tavasz and ősz - spring and autumn, respectively. The suffix in question is -val/-vel, which is usually translated as "with". The v of the -val suffix gets assimilated to consonants, so if the last letter of the base noun is a consonant, it gets doubled. Some examples:
- ház - házzal (with a house)
- a barátom - a barátommal (with my friend)
- a kutya - a kutyával (with the dog)
- repülőgép - repülőgéppel (with an airplane / by airplane)
Now, why do you say tavasszal and ősszel, but nyáron and télen? I have no idea, but it sounds weird the other way around. A couple of time-related words behave that way, for instance you have
- nappal - in the day
- éjjel - in the night
- reggel - in the morning (the base word reg is not used anymore)
- este - in the evening
- délben - at noon
- az évben / az éven - in the year
- a hónapban - in the month
- a héten - in the week
- az órában - in the hour
- a percben - in the minute
Expressions of time are one of the most difficult things to get right in about any language, so you'll just have to get used to it, I guess. There's often no good reasoning behind it. Like if you say in English you're doing something "in a week" if it's actually after a week.
I think this is good information for this stage- Duolingo are just stretching us by bringing in more at each point than where we might actually be- but when we come across it later on, it will have a familiarity to it that will help a lot. Duo are also giving us larger chunks of language than just "spring" and "autumn", so that when we come to use them, we will have a better chance of getting the right suffix and be close to making a good sentence! Maybe there's another section on it later on as well?
Tavasz, nyár, ősz és tél. - Spring, summer, autumn, and winter.
The words with the -i at the end are the respective adjectives. Autumnal, wintery, and so on.
The suffix here is -val/-vel, usually translated as "with", but can also occasionally be used with words of time to mean "in". It's a bit a tricky suffix, because if the base word ends with a consonant, the v of the suffix gets assimmilated:
- a kutya - a kutyával (with the dog, all is normal)
- a barát - a baráttal (with the friend. v -> t)
- banán - banánnal (with banana. v -> n)
- ősz - ősszel (in autumn. v -> sz, making a double-sz which is written as 'ssz')