Partnership
You don’t need a formal agreement to enter into partnership (aka “partnership at will”). Agreements are useful if a complicated business or in the case of disputes.
Partnerships are easy: you and mate agree how you’ll split the income. Each of you gets your share. You’re taxed as a self employed person (i.e. normal income tax).
Easy to administer – no accounts required. Nothing has to be filed at companies house.
But, they don’t protect you if there’s a claim against you. So, if your mate causes a client loss, they can sue him, him and you or just you.
Limited liability partnerships
These are a hybrid of a company and a traditional partnership. You are not partners, but members.
More like companies – limited liability. Have to file information at companies house (including accounts, which will have to be audited if over a certain income). More administratice burden.
Still tax “transparent” – i.e. you just pay income tax (and NICs) on your share of the LLP’s income.
Company
Easily identified – name ends in “limited” or “plc”.
You want a private limited company (“limited”). This, like an LLP, has a separate legal identity. You become shareholders (if limited by shares) or members (if limited by guarantee. Profit making companies are usually limited by shares.
You hold the number of shares to give you your ownership proportion.
Company receives income and pays corporation tax on it (depending on thresholds). You receive money out of company either as salary (PAYE + employer & employee NICs) or as dividend (not PAYE, no NICs).
You also need directors (can be same as shareholders), and need to jump through various admin hoops (companies house filings, accounts, etc.).
Easy..!