Strain: injury that affects muscles or tendons that attach muscles and bones. These occur after a quick tear, twist, or pull. You can get these after overstretching or an over contraction. With a strain, you can experience pain, weakness, and muscle spasms.
Sprains: injury that affects ligaments, which is the cartilage that attaches bone to bone. You get a sprain when you stretch or tear the ligament. Usually occurs after trauma (fall or something that will displace the joint from the normal alignment). Can be mild tear or complete. You will usually experience bruising, swelling, instability, and painful movement.
Regardless of what you have, the treatment is the same. RICE: Rest, Ice, Compression, Elevation. Obviously, rest the injured area. Ice for the first 24-48 hours to prevent further swelling and damage. (Only apply for 20 minutes at a time). Compression, which is in the form ace wrap or splints. And then Elevation. Elevate the injury extremity above the level of the heart. This helps if your foot is swollen. This will help mobilize the swelling from the foot and pull it back up your leg. This takes the pressure off of your foot and helps reduce the pain. Also, if you are continuing to have pain, you can take NSAIDS (Non Steroidal Anti-inflammatory Drugs. These are over the counter, such as Advil or ibuprofen.)
Tips to avoid getting this injuries:
Avoid exercising or playing sports when tired or in pain.
Eat a well-balanced diet to keep muscles strong.
Maintain a healthy weight.
Try to avoid falling (for example, put sand or salt on icy spots on your front steps or sidewalks).
Wear shoes that fit well.
Get new shoes if the heel wears down on one side.
Exercise every day.
Be in proper physical condition to play a sport.
Warm up and stretch before playing a sport.
Wear protective equipment when playing.
Run on flat surfaces.
These injuries usually heal after a week or two depending on how bad the strain/sprain is. If these is a complete tear, that would require surgery and you would need to be seen by an orthopedic doctor.
In regards to answering your first question, I am not completely comfortable diagnosing whether you have a strain or a sprain over the internet without doing my own physical exam on you. But, since haven't mentioned any instability, I would think that you would have a strain.