There are two kinds of Publishing deal: a single-song assignment and an exclusive agreement.
A single-song assignment, not surprisingly, grants the copyright in one specific song to the Music Publisher. The royalty split varies between 50/50 (most common in North America) to anything up to 100/0 in favour of the writer: the latter is extremely rare but does exist where the income is so high that the Publisher is happy to merely earn interest on the total sum before paying the writer. Advances are usually minimal for a single-song deal.
The Exclusive agreement ties the songwriter to the Publisher for a certain period, usually from one year to the next, and states that every song written or co-written during that period is automatically assigned to the Publisher. This is sometimes referred to as a "staff" deal as it has many of the advantages associated with a "normal" job, such as a regular salary, place of work, etc. The salary one could expect would not usually be too extravagant but it will be enough to live on.