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A consultancy agreement allows a client to contract a consultant for services or a deliverable which will be defined in the statement of work. A consultant is not an employee of the client.
A fixed-term consultant is usually hired for a specific project or deliverable which won't need the consultant's support once they have been delivered.
SEO consultants help a client improve the findability of their websites on search engines. SEO consultants will will usually start with an audit of the website before suggesting and implementing an SEO strategy and improvements.
An SEO consultant will usually be exposed to the client's strategic information during the project which is why the consultancy agreement should have confidentiality provisions to provent them from sharing it with competitors or disclosing it to the public.
It can take many months until SEO initiatives start to bare results which is why it can be cost effective for a client to contract an SEO consultant in the early days to receive strategic input and launch initiatives before hiring a full-time SEO lead.