Hi Sandra! It seems that you are attempting a formidable task, and more of a detection job than plain old genealogy. It's difficult to offer specific advice without knowing what facts you already know, and also what you may be able to surmise. I'm guessing that your mother is no longer around, or is not willing to co-operate with your search. Are there any siblings or friends of your mother who might be able to help?
Finding an unknown father isn't possible by using genealogy alone, unless his identity was recorded somewhere, perhaps on a birth certificate or adoption papers. If he is actually unknown to everyone except your mother then the first action to take is to gather up a list of possible subjects. It's not unusual for the man himself to be unaware that he is a father, and in some cases even the mother isn't certain which of the men she had relationships with was the father of her child. But a little background study of the social groups in which your mother moved in the year prior to your birth should enable you to identify some likely men.
I would search for men in an age range from one year younger to five years older than your mother,in the following groups (and any others that you might think of) - neighbours, workmates, attendees of the same schools/colleges as your mother, attendees of the same church as your mother. Compiling this list is likely to show up some patterns in your mother's life, which might then give you more ideas of the best direction to head your search. When you have a list, pick out those individuals who appear more likely possible fathers, and find out everything that you can about them - what they were doing around the time of your conception, and what they did in the time immediately afterward.
If you can find some men who seem strongly likely to be your father, then you need first to ascertain whether they are still alive, and if not, whether they have descendants. You should then consider very carefully whether to approach them, and how best to do it. Once contact is made, first ask if they believe it possible that the individual is your father, and then ask if they would be willing to undertake a DNA test for comparison.
Good luck with your research!