I applied for obamacare around the end of jan or middle of feb .. how long does it for my application to process, or receive the insurance cards?
Applying and enrolling aren't the same thing. For most people the application process takes 15 minutes. Getting the application approved can take longer depending upon your situation. Once you are approved, you'll still have to enroll in a plan.
If you enrolled in a plan before the 15th, the plan starts on the 1st of the next month after you pay your first months premium. If not it starts the month after next month on the 1st.
If your plan hasn't started yet, you may not have finished the enrollment process. Go and call healthcare.gov now to ensure you finish the enrollment process ASAP!