SharePoint actually comes with a few built in ways to do this!
The first way is simpler:
in the Document Library Settings > Versioning Settings you can enable content approval to be required where anyone with approval level permissions can approve the document from the document library using the ... button. To acquire the emailing task notification and some other more advanced features you will need to use the workflow.
To use the ready workflow you have to enable the "workflow feature" for the site collection (top level site settings > site collection features) which then adds a set of "Aggregated of out-of-box workflow features" one of which is the approval workflow.
To use this workflow go to the Document Library Settings > Workflow Settings > Add Workflow and the workflow template will be the approval - SharePoint 2010. The task list is then where the list of tasks can be seen with info like who the task has been given too and status of the workflow. You can also set it to run when a document is created or manually . The basic structure of this work is as follows:
- Collect data about approval (deadline, approver etc.)
- Send email to the approver (with link to document and to task list)
- The approver then approves/rejects the document either from viewing the document or through the task list
- The creater of the email is then sent an email with whether it has been approved or declined
The Microsoft Website is really good at explaining the workflow if you need more information and if you need to expand it's capabilities SharePoint Designer can be used to customize.