A Seed Round is likely to be a company's first formal funding round. Before the Seed Round, the company might have taken pre-seed investment as one-off sums, or as a Bootstrap Round or via crowdfunding.
Series A is the funding round that follows a Seed Round. A company might raise Series A financing to grow the business, to continue developing the products or service, and to hire more people.
Like a Seed Round, Series A is equity-based so the company receives money from investors in exchange for shares in the company.
After Series A, if the company needs to continue fundraising, the next round would be Series B, followed by Series C and so on.