When two or more people go into business together, they've automatically formed a partnership; they don't need to file any formal paperwork. By contrast, to form a limited liability company (LLC), business owners must file formal articles of organization (sometimes called a certificate of organization) with their state's LLC filing office (usually the secretary of state or department of corporations) and comply with other state filing requirements.
Aside from formation requirements, the main difference between a partnership and an LLC is that partners are personally liable for any business debts of the partnership -- meaning that creditors of the partnership can go after the partners' personal assets -- while members (owners) of an LLC are not personally liable for the company's debts and liabilities.
There is one similarity between LLCs and partnerships, however. They both offer "pass-through" taxation, which means that the owners report business income or losses on their individual tax returns; the partnership or LLC itself does not pay taxes.