There is a "typical" set of steps that takes place, but this vague framework varies quite significantly in duration and content depending on project complexity, scale and negotiation duration.
The basics are as follows:
The client requests a proposal. This request may be made (1) publicly, or (2) to a group of companies or (3) to one company in particular. A company in this context also may be an individual person offering a service. This request will consist of a description of the job to be performed and varies wildly in length, from a few sentences to pages.
The company (aka contractor) responds by submitting a proposal describing how the job will be done. The company may ask questions about specific details of the project if they need more information. It may be necessary for the contractor to sign a Non-disclosure agreement (NDA) in order to get the information they need. There may be a back and forth for a while between the client and contractor. This stage allows for the client to asses the contractor and for the contractor to asses the client. The proposal may contain a cost estimate.
At this point, it's a good idea to establish a scope for the project, which is a definition of exactly what the final deliverable consists of (and what it does not consist of!)
The client will ask for a quote if they like the contractor and want to move ahead. If they do, the contractor sends through a formal quote, detailing exact amounts to be paid and when.
The client decides if they're okay with the quote. Once again, negotiation may occur over the price of various services and milestones. The scope may be revised to make things cheaper (and re-quoted). If they're okay, the contractor issues an invoice for the first installment. This invoice is what the client needs to pay for the work to start
The client may send through a contract to the contractor detailing the legal responsibilities of each party. This in turn may consist of several stages of negotiation in order to reach an agreement over the contents of the contract. Both parties sign.
Once the client pays the invoice, the contractor issues a receipt to the client, confirming the receipt of payment. The work begins.
At this point, the documentation supplied is dependant on the scope and what documentation was agreed upon as a deliverable.
That is a vague summary of the business transaction process. Since there is a lot of negotiation about price and scope and NDAs, it can vary in duration from hours to weeks. It may be as simple as a client emailing a contractor about a small website change or as complex as a government municipality subcontracting the design and construction of a new stadium.
(1) Request for proposal: Description of job from client.
(2) Proposal: Description of how the job will be done from contractor.
(3) Estimate: Estimate project cost.
(4) NDA: Non-disclosure agreement ensuring contractor doesn't give client's IP away.
(5) Scope: Description of work to be done (and what's not being done).
(6) Quote: Actual amount to be paid at each stage and terms of payment.
(7) Contract: Legal document detailing responsibilities of each party.
(8) Invoice: Current amount to be paid to the contractor.
(9) Receipt: Confirms that payment was received for a particular invoice.