A partnership is a separate legal entity from a sole trader. It requires two or more individuals to carry on business together. You could have a partnership with your spouse, your children if not minors, and any unrelated parties.
Partners are taxed on their share of the profit arising to the partnership. This is calculated in the same way and subject to the same rules as for Sole Traders.
A partnership with your spouse can be useful for tax purposes for saving income tax under the individualization rules and for Capital Gains Tax purposes under the retirement relief rules.