As mentioned above, accounting profit is the surplus we can find on a balance sheet. It can be calculated as the difference between total revenue and costs. However, the important aspect here is that accounting profit only includes explicit costs. That is, it only accounts for costs that result in an outflow of money (or an increase in dept) for the firm.
For example, think of an ice cream seller who wants to open a new business. Let's assume he faces costs of $100'000 for equipment and ingredients. At the end of the year, he has sold ice cream for a total of $150'000. Thus he makes an accounting profit of $50'000 ($150'000 - $100'000).
This approach is rather business-oriented. It includes everything that is relevant to set up a balance sheet. However, since we are looking at the issue from an economic perspective, we need to include some additional aspects. Therefore, we shall look at economic profits.
Economic profit is defined as total revenue minus total costs. That means in addition to the explicit costs it also includes implicit costs, such as opportunity costs. In other words, economic profit also accounts for the time, money and effort an owner puts into his company.
Regarding the zero profit condition, this suggests that in the long run equilibrium, owners need to be compensated for their opportunity costs. Hence the company must actually generate a positive accounting profit in the amount of the opportunity costs incurred. Therefore it will generate at least the amount of profit that is needed to maintain the factors of production (labor, capital, etc.). This profit is referred to as the normal profit.
To illustrate that idea, let's go back to our ice cream seller. If he had decided not to set up his business, he could have for instance deposited the $100'000 (i.e. the explicit costs) in a bank account to earn $5'000 in interest. In addition to that he could have worked for another ice cream producer to earn $45'000 a year. As a result, his opportunity costs add up to $50'000. If we include those costs in the calculation of the accounting profit above we get the economic profit for this case, which will amount to zero ($150'000 - $100'000 - $50'000).
This shows that even if economic profits are zero, producers still earn positive accounting profits. They have no reason to go out of business, because they receive compensation for their opportunity costs, so there is no alternative that would generate higher profits for them.