Synchronizing with network – What does it mean? The messages:

synchronizing with network   
xxx blocks remaining


what does that mean?
It's most likely you're using the client from bitcoin.org.

Synchronizing with the network refers to the download and verification of the blocks in the block chain, which is the transaction history of Bitcoin. The block chain is the record of who spent what and when they spent it, and who owns what now.

Currently, the bitcoin.org client downloads the entire block chain on installation, which is somewhere around 3GB of data. As it downloads each block, it confirms each transaction, which is a disk-intensive activity. Unless you have high spec hardware, expect it to take many hours, possibly even days.

Version 0.8 of the bitcoin.org client alters the synchronization process, which heavily reduces the time required to get the client functional. The process is faster through being less disk intensive.

Alternative clients are also available, such as Armory or Electrum, which don't use the same synchronization process. Electrum doesn't download the block chain, and the features list for Armory says it doesn't need the block chain.