You haven't mentioned which version of SharePoint you are running. I presume because you don't have the option available you are running SharePoint 2013 Foundation.
SharePoint 2013 Foundation doesn't use support the Workflow Manager and uses the same workflow engine that SharePoint 2010 Foundation uses- this doesn't include the approval workflow.
The Approval-SharePoint 2010 workflow template comes with SharePoint Standard or SharePoint Enterprise. See here for feature comparisons between versions:
https://blog.blksthl.com/2013/01/14/sharepoint-2013-feature-comparison-chart-all-editions/
Unfortunately, having tried various things myself, the options are upgrade to at least SharePoint standard or build a custom workflow. Customer workflows can be built using SharePoint Designer or as part of a code solution. There are quite a few guides already out there. This is just the first one I came to:
http://plexhosted.com/billing/knowledgebase/226/How-to-create-a-simple-approval-workflow-in-SharePoint-Designer.html
If I am being too presumptuous and your version is higher than Foundation then it might just be that you need to activate the Workflows feature in the site collection features.