The simplest miner would work like this:

1. Request a new Getwork from a given pool/bitcoind with specified credentials
2. For nonce=0;nonce<0xFFFFFFFF;nonce++
   • Set Getwork Data nonce
   • SHA-256 hash the Getwork Data (block header)
   • Check if hash result is smaller than Getwork Target, if so, submit a share
   • If more than 1 second passed since you received Getwork, stop the loop
3. GOTO: 1

Now, if you want something that could rise up to par with currently used miners, you need to consider:

• OpenCL calculations, without this you will never get the high hashrates of graphic cards.
• Midstate, or in other words hashing optimization. First half of Getwork Data doesn't change, so no point in calculating midstate for each hash.
• Longpolling, makes you calculate less stale blocks.
• Rollntime, minimizes communication with the pool.

Read the answers to this question and if you have any specific questions about the mining algorithm perhaps you can edit your question to ask them.

If all you want to do is run a mining program, there are lots available. You can study the source and see what they're doing.

The official bitcoin client itself has a basic CPU miner built in. You can examine the code to get an idea of how mining works.