I am currently doing a project that requires blockchain to be applied to other systems. I was wondering where I can easily download the blockchain database and which is the best way of understanding blockchain.
A block chain is a transaction database shared by all nodes participating in a system based on the Bitcoin protocol. A full copy of a currency's block chain contains every transaction ever executed in the currency. With this information, one can find out how much value belonged to each address at any point in history.
If you are new to Bitcoin or have never worked with 'bitcoind' (the Bitcoin daemon with no user interface) then I suggest you start clicking (and reading) around on these sites:
en.bitcoin.it - A Bitcoin wiki
bitcointalk.org - A Bitcoin Forum
Unless one really dives into the source code of Bitcoin one can't really manipulate (or sometimes even view) a Bitcoin blockchain without running the Bitcoin software.
Most blockchain software is running a form of original Bitcoin software underneath the blockchain software.
If you are looking to try and do something like blockchain.info then I would suggest Abe to start with. Abe draws inspiration from Bitcoin Block Explorer (BBE) and BlockChain.info and seeks some level of compatibility with them but uses a completely new implementation.
If you are looking at Abe like it could be way over your head then try googling "bitcoin blocknotify" or "bitcoin walletnotify".
Block and wallet (and alert) notify is a parameter you can set in your bitcoin.conf file that will make your bitcoin wallet execute a script of your choosing and can pass the script an argument.