Lel. No problem. Basically,
Constructed = Ranked and casual play, but mostly just ranked. Casual is usually where decks are tested and tried in. You'll also see a lot of beginners playing there. It's best to start playing Ranked as soon as possible, though, so you can at least get to rank 20. You can't go lower in the ladder once you reach 20, and you start at 25. Getting to rank 20 in a season gives you like a card back or something, and the highest rank you get in a season gives you better prizes. I think it's always golden cards, improved in rarity every 5 ranks, and additionally a little bit more dust every individual rank. Dust is what's used to craft cards.
Arena = Waaaay different and more complicated than constructed. You have to pay 150 gold (or like $1.50) to get in, but the first one is free so it's definitely worth doing for the free rewards you get just for losing participating. Basically, you get a choice of three random classes from the nine total, and from there you get to pick one card out of three of the same rarity (common, rare, epic, legendary), and you pick 30 in total. Obviously, the rarer cards are, the less likely you are to get to draft them. It gets really complicated because not only do you have to worry about the draft (you picking your cards), which includes balancing minions/spells based on your hero, making sure your mana curve is really flexible, and ensuring you'll have the best play available for every turn, but you also have to make sure your deck is more or less ready for the different levels of arena. There's a maximum of 12 wins in a single arena run, and it ends immediately after your third loss. Obviously, the rewards at 12 wins are higher than the rewards at say, 4. So you have to end up worrying about 1) getting a good hero for arena, 2) getting a good draft that doesn't fall short against the lower-leveled zoo decks (basically tons of minion spam) while managing to hold up later against the much better/luckier decks/players, and 3) making sure you're not wasting turns wherever possible by always playing something that gives you the advantage/board control/pressure on opponent.
It sounds way more complicated than it needs to be, and that's because it is, but it's not that tricky when you get into it. Some people just about exclusively play arena. After I think 4 wins, you have a 100% chance to get enough gold for another arena run, so people who go 12-wins several times in a few runs can easily afford more arena and more packs. If you want to get good cards and be a good player, go for arena, but make sure you know what you're doing beforehand. From experience, I can tell you, a bunch of 0-3's, 1-3's, 2-3's, and 3-3's are not worth the potential many packs that cost about the same when you so crucially need them for constructed.
Also: if you don't want to put any money in, I'd recommend dusting/disenchanting golden cards that are often viewed as unusable trash. You get more dust for disenchanting a golden card than its regular counterpart, but the amount to disenchant a golden card is always equal to the amount of dust it takes to craft the regular counterpart. Basically, you can't disenchant a golden card, craft its regular version, and then have more dust from before you started.
^This way you can bypass any shitty luck and get super important cards like Dr. Boom, Sylvanas Windrunner, and Alexstrasza. They're very good for most builds, particularly the first two, and open up a lot of doors.
Yep. Official FSE Hearthstone player.