I am currently spending a lot of time researching this out, and I'm finding that there are not very many options out there.
Cross Topic:  https://bitcointalk.org/index.php?topic=22079.0

Current Options:

1.  Run a bitcoin Full node.  Write a program to interact with the JSON RPC API.  
(note that the https://github.com/freewil/node-bitcoin code mentioned by coinableS is actually not an implementation of address accounting software, but a way to communicate that simplifies the JSON-RPC interface a little bit)
Also, I seen this software that doesn't seem to be maintained, but looks like a good start on address management:  https://bitcointalk.org/index.php?topic=408465.0

2.  Use a third Party API such as:     (However, even using a third party api, you should choose to manage your bitcoin private keys yourself, and which user/account these addresses are associated with.)  The API should only used for tracking/sending  payments to addresses. (The transaction should be signed with the address private key using api scripts that run locally on the Application server.  For greater security, the private keys should not be stored on the application server unencrypted, or not at all, because the transactions could be queued, and sent to another more private server for transaction processing.

- http://blockchain.info/: Well of course we all know this one.  Bitcoin only, access is via custom API.   (note the limitation of addresses given by coinableS)
- https://apicoin.io/: "Put the Bitcoin node in the Cloud!"  Bitcoin only, access is via custom API.
- http://www.blockcypher.com/: "Powering block chains in the cloud".  Bitcoin and Litecoin, homepage suggests they may add new coins, access is via custom unified API.
- https://helloblock.io/: "Focus on building Bitcoin apps, not infrastructure."  Bitcoin only, access is via custom API.
- https://chain.com/: "The Block Chain API for Developers"  Bitcoin only, access is via custom API.
- http://blockr.io/: BTC, LTC, TBTC, DGC, QRK, PPC, MEC, access is via custom unified API.
- https://www.blocktrail.com
- http://www.bitcoinmonitor.net/


Third Party API Topic:  https://bitcointalk.org/index.php?topic=655722.0

Something to be aware of when generating addresses for each user with blockchain.info is the wallet will "max out" at 1,000 addresses. They don't tell you this upfront, which is kind of lame if you chose this option for your site only to find out later. Maybe you want to generate the address and keys yourself and then assign the addresses to each new user as they sign up. You can use blockchain.info API for checking balances or using a hot wallet for sending payments.

There's also several options with running bitcoind on a VPS and then use jsonRPC calls.

https://github.com/freewil/node-bitcoin