How to approve or reject the record without login to salesforce?

Go to Setup --> Build --> Workflow & Approvals --> Process Automation Settings --> Check "Enable Email Approval Response" option --> Save.

This option allows, user can approve/reject record without salesforce login.

Enabling email approval response lets users reply to email approval requests by typing APPROVE or REJECT in the first line and adding comments in the second line.

Possible email response reply for approving records,

1. APPROVE
2. APPROVED
3. YES

Possible email response reply for rejecting record,

1. REJECT
2. REJECTED
3. NO

If replying via email, you can also add comments on the second line. Comments will be stored with the approval request in Salesforce CRM.

Reference:

Enable Email Approval Response