For instance, if someone sues me because they do not like the product offered or if someone else sues me because of a contract violation. How much is the owners responsible for (in this case me) and how much is the business responsible for? Or, in the eyes of legal matters, is the owner of the partnership or proprietorship responsible for all legal matters?
Suits against sole proprietorships and partnerships can access the owner's (owners') personal assets directly - so yes, you are liable for damages from a contract violation. This is why many businesses opt to go with LLCs, LLPs, Corporations, and other limited liability entities.
Keep in mind that even setting up an LLC or corporation isn't necessarily the way to get certain protection.
To protect yourself, form an LLC or a corporation AND:
Google "pierce the corporate veil" and there will be more articles on this.