There are various ways that an F-1 student might be eligible to work and/or to get an Employment Authorization Document (EAD) - the best bet is to consult your school's DSO for more information.
If when you say "completion of course" you mean actual graduation from the program rather than just a specific course, then what you likely refer to is something called "Optional Practical Training" or "OPT." This does require an EAD, applied for through USCIS after an endorsement on your I-20 by the DSO. You must be working in a field directly related to your studies.
You can get up to a year initially on OPT - but reduced by the amount of time spent on practical training BEFORE graduation (you get a year total of practical training, and this can be split between during-school and after-graduation).
There are a couple of modifications to this general, one-year rule: if you are working for an employer, and the employer is happy with you and chooses to sponsor you for an H-1B visa which likely won't begin until October 1, the OPT can be extended up to four more months to reach that date since most tend to expire over the summer.
Also, certain people can extend the OPT up to 29 months total - but only if their degree is in a Sciences, Technology, Engineering or Mathematics ("STEM") field, AND the company with which they are working is enrolled in a government employment-authorization-verification system called "E-Verify."
Again, consult your DSO for employment authorization inquiries - and consult an immigration lawyer for more information on H-1Bs and future plans.