After you completed installing Bitcoin client you need to introduce yourself with some things that will keep your Bitcoin faster and clear out some misunderstanding about money transactions. You should increase size of database cache to decrease your computer disk usage and increase syncing speed. You may need to add -txindex option to command line to study Bitcoin transactions futher and to use Bitcoin in your business.

You can send bitcoins to others with transactions. Transactions are digitally signed agreements with your private key that contains your will to transfer money and proof based on your transaction history that you have received some bitcoins that are not already spent. Transactions are identified by transaction identification number aka transaction ID. Transaction ID depends on transaction contents but there is more than one way how signature is included in transaction so transaction ID can get different. You should not rely on transaction ID. You should retrieve your transaction with getrawtransaction command. If your transaction looks stuck or lost you can resend it with sendrawtransaction (use output from getrawtransaction command). If after that you see errors like inputs already spent you may use listtransactions or listsinceblock commands to retrieve transaction with changed TxID. Read more about Transaction Malleability.

Transactions are confirmed by group of users called miners. They combine transactions all together and solve cryptographic problem, this process is called mining (more details). These transactions and problem solution are then combined together and this is called block. Block identification number aka block hash depends on block contents keeping content intamperable without changing block hash.

Block also contains one previous block hash. You can imagine now these blocks are coming one by one forming chain of blocks - blockchain. But there is problem - two participating miners can create blocks based on one block. Only one will get in main blockchain, best block.

Your transaction may get lost in orphan block or looking invalidated because your transaction input (also transaction) got included in block with different transaction ID. So there can not be absolutely 100% confirmed transaction, it is confirmed by blocks mined over it. More confirmations - less probability TxID will change or transaction reverted. You should not repeat MtGox faults.

One more thing. Your transaction may get stuck if you do not put enough fee for miners. It is safe to resend transaction that contains higher fees. Just include at least one input from stuck transaction to make sure only one of them gets mined.

All other Bitcoin knowledge is up to you. :)