If a speaker has two crossovers, then the speaker likely contains 3 different drivers. The woofer, the mid-range and the tweeter.
The crossover indicates roughly where one speaker driver stops "working" and the next takes over. So, in your example, the tweeter would take over from the mid-range driver at 2kHz. The drivers don't actually "stop" working since there's a rolloff in the frequency response, so there are typically electronics involved at the crossover.
Of course there are speakers with only one or no crossovers depending on the number/type of drivers. Some higher priced speakers may contain even more drivers - have a number of woofers, tweeters, etc.
Then there are subwoofers, that take over from the regular speakers at the low end. These crossovers are usually handed by the AV Receiver, although not always, depending on your configuration.
Sam you pretty much have that spot on. Now a days the subs crossover can be adjusted by the user either in the Amp set up or on the sub itself. I personally set the sub crossover in the amp to as high as possible or pass-through and then adjust the crossover on the actual sub.
Although most speakers are capable of reaching a range of 30 or 40hz I always have my sub come in around 60 or 70hz. That way you can set your speaker crossovers in your amp up higher so you aren't working your amp(s) and speakers as hard and are allowing your sub to produce the low end you are looking for.
I usually set my crossover to the sub (on the sub like the poster above) around 80 hz. My suggestion is play with the settings and see which one sounds good to you. There's the science part and there's the "sounds-good-to-me" part.
Then with your favorite beverage in hand, enjoy the fruits of your hard work.
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