Understanding Bitcoin wallets I want to make sure I'm understanding a few basic concepts clearly:


A Bitcoin wallet itself is nothing more than a "record" that is stored on the Bitcoin blockchain. It's an identifier that will be used in a person's Bitcoin transactions.
A Bitcoin wallet app is simply a software tool to create transactions that are written on the Bitcoin blockchain
I think some Bitcoin wallet providers -- not sure if "provider" is the right term -- may keep some information about the user's wallet/transaction information on their own system.
The value added by those who provide Bitcoin services are simply providing user-friendly tools to read/write information from the Bitcoin blockchain. They may also be providing a method of converting real currency into cryptocurrency and vice versa (e.g., trading or purchasing of Bitcoin)


Am I getting this right so far?
Not sure if this is helpful, but here's what I think you should think about when you hear "Bitcoin wallet":


A Bitcoin wallet is a set of accounts that you can (a) spend coins from and (b) receive coins into.
An account is one and the same thing as a secret key and its corresponding public key [1].
To spend coins from an account, you need to perform a digital signature using that account's secret key.
Thus, anybody in possession of the account's secret key can spend all coins from that account. In other words, your secret key better stay secret and please don't lose or "forget" it. Or else [2].
To receive coins from Alice, you have to give Alice one of your accounts' public key [1].


Now, armed with this knowledge, I think you can answer your own questions:


  A Bitcoin wallet itself is nothing more than a "record" that is stored
  on Blockchain. It's an identifier that will be used in a person's
  Bitcoin transactions.


Yes and no. A wallet is a bunch of accounts. Each account has a "record" in the blockchain: there will be an entry of the form (_public key_, 10 bitcoins) for that account [3].


  A Bitcoin wallet app is simply a software tool to create transactions
  that are written on BlockChain


Yes. A wallet app is a piece of software that manages all of your accounts' secret keys and enables you to perform those digital signatures that let you spend your coins. Those signatures are over transactions of the form (_my public key_, _recipient public key_, x bitcoins) as I detailed in [3].


  I think some BitCoin wallet providers -- not sure if "provider" is the
  right term -- may keep some information about the user's
  wallet/transaction information on their own system.


I'm assuming you are talking about folks like coinbase.com, which manage your wallet for you? Recall that if they're gonna allow you to spend your bitcoins with a click of a button online, then they need your accounts' secret keys. So yes, they store that. And sometimes they're hacked and you lose your money [2]. On the other hand, if you don't use online wallets like coinbase.com, managing your own secret keys can be dangerous too: I have heard many stories of people losing their hard drives and thus their bitcoins.


  The value added by those who provide BitCoin services are simply
  providing user-friendly tools to read/write information from
  Blockchain. They may also be providing a method of converting real
  currency into cryptocurrency and vice versa e.g. trading or purchasing
  of BitCoin


The value part of this is in "opinion land", but yes, I suppose I agree: it's much more user-friendly not to mange your secret key and worry about losing it or being hacked. Still, you might just be replacing one worry with another: now coinbase.com might be hacked and your bitcoins gone [2].

And yes, some wallet providers like coinbase.com let you convert USDs into bitcoins and viceversa.

I hope this helps.

[1] Sometimes you'll hear the term address, which can be thought of as a "fingerprint" of the public key. To keep things simple, I will not distinguish between the two. Although, in practice, you will give the address, not the public key, to folks who want to send you coins. (This has to do with efficiency reasons and a bit of security.)

[2] Every time you hear about one of these "Bitcoin hacks," it's because someone's secret key got stolen =&gt; the victim's coins got sent to the attacker's account using a digital signature via the stolen secret key.

[3] To be more specific, there are actually a bunch of entries recording transfers from some other account into your account: (_public key_ of sender, your account's _public key_, 5 bitcoins sent to you)