The legal form of a business decides three things: who is responsible for its debts, how its profit is taxed, and how its owners take money out.
Sole proprietorship
One owner, no separation between the person and the business. It requires no formation and is the default for anyone trading without setting anything up.
The owner is personally liable for everything the business owes. Profit is taxed as the owner's income and carries self-employment tax on all of it. Money is taken out as an owner's draw.
Partnership
The same arrangement with more than one owner. Profit is divided by the agreement between them and taxed to each individually.
Without a written agreement, the default rules of the jurisdiction apply, which are rarely what the partners would have chosen.
Limited liability company
A separate legal entity. Its debts are its own, which is the point of it: personal assets are generally protected from business creditors.
For tax it is flexible. By default a single-owner company is taxed like a sole proprietorship and a multi-owner one like a partnership, but it may elect to be taxed as a corporation instead.
That protection depends on keeping the business genuinely separate. Paying personal costs from the business account undermines it — see marking a transaction personal.
Corporation taxed as a pass-through
A corporation that elects to pass its profit through to its owners rather than be taxed on it directly.
The reason owner-operators choose it is the split between wages and distributions: only the wage carries payroll tax. That split is constrained by reasonable compensation, which exists precisely to stop it being abused.
Corporation taxed in its own right
The company pays tax on its profit, and shareholders pay again on what is distributed to them.
Common for businesses raising outside investment, uncommon for owner-operated service businesses.
Choosing between them
The trade-off is administrative burden against tax treatment and liability protection, and it turns on figures specific to the business. It is a decision for an accountant and an attorney who know it.
The name a business forms under and the names it trades under are a separate filing — see business name registration.