In economic terms, a normal profit is the profit over and above a normal profit. A normal profit is also defined as zero economic profit. This is when the firm's revenues exactly equal its explicit and its implicit costs.
Explicit costs are the costs that a firm actually pays out. This includes things like salaries paid to its employees and the costs of the materialis it uses.
Implicit costs are the opportunity costs incurred by the owners of a company. This is the value of the other things they could do with their resources instead of using them for the business of this particular firm. For example, if you have a job making $50,000 per year and you quit it to start your own company, you have implicit costs of $50,000 because you could be making that much if you weren't running your own company.
When a firm's revenues exactly match its explict and implicit costs, it makes a normal profit. Anything above that can be called "economic profit" or "abnormal profit."