Bitcoin is a currency built ontop of cryptography and computing power it relies on the blockchain, a distrubted ledger to allow for low cost transactions (much much lower than that of any banks).

As a developer you should know that this is likely to become a successful payment method especially where digital goods are concerened. however you are unlikely to need to learn about the blockchain or cryptography etc to be able to handle those payments or to begin to start interfacing your existing services with bitcoin services. the process for us will be similar to how you would integrate paypal or stripe [ unless of course you ar insistant on taking the repsonsibilities yourself in which case you can use the official bitcoin software and some prebuilt libraries to buil your own services (in this cae you might have to learn about the blockchain and cryptogrpahy)]

Be wary if you are going to be paid in Bitcoin as it has a volatile price (time of writing at 352 EUR per bitcoin within the last 6 months we have seen 200EUR and lower, and also as high 450EUR), bitcoin is considered anonymous by some (it is not completely anonymous, although wallets are not connected to users so finding the owner of a wallet could be very diffuclt ), but it can be hard to track down after a few transactions when the trail gets 'muddied'. bitcoin has many PROs and many CONs I personally beleive it has a great future and embrace it, but only time will tell.

Bitcoin price API's can make an excellent source for data-vis/d3.JS projects, if you are looking for some interesting or arbitrary data to practice some SVG manipulation etc!

Hashing algorithms :-)

In ultra short words. Block-chains of hashes.

Think of it like an action log, where each log entry contains the hash value of the preceding log entry. By walking the whole block chain, you can/must validate that each hash value matches the content of the preceding log entry (you do that usually only once and then stay current with recent hashes). To write into the log, you need to be fast or you are forced to wait and hash & revalidate the latest log entries again. This is similar how Ethernet works except that each client doesn't get notified about collisions.

Because of all those linked hashes, it's pretty hard to inject or manipulate history data. But it's also very computation heavy.

As a developer, knowing about Bitcoin may not be required, but learning about Blockchains (the technology on which Bitcoin is built) will stand you in good stead.

Search for blockchain tutorial or blockchain online demo for some good resources.