<?php

$username = $_POST["username"];
$password = $_POST["password"];
$institution = $_POST["institution"];

$conn = pg_connect("database connection information"); //in reality this has been filled
$result = pg_query($conn, "INSERT INTO institutions (i_id, name) VALUES (null, '$institution') RETURNING i_id");
$insert_row = pg_fetch_row($result);
$insti_id = $insert_row[0];

// INSTITUTION SAVED AND HAS ITS OWN ID BUT NO MEMBER OF STAFF ID

$resultTwo = pg_query($conn, "INSERT INTO staff VALUES (NULL, '$username', '$password', '$insti_id'");
$insert_rowTwo = pg_fetch_row($resultTwo);
$user_id = $insert_rowTwo[0];
// USER SAVED WITH OWN ID AND COMPANY ID
// ASSIGN AN INSTITUTION TO A STAFF MEMBER IF THE STAFF'S $company_id MATCHES THAT OF THE
// INSTITUION IN QUESTION
$update = pg_query($conn, "UPDATE institutions SET u_id = '$user_id' WHERE i_id = '$insti_id'");

pg_close($conn);

?>

CREATE TABLE institutions(i_id serial, name text, u_id int);
CREATE TABLE staff(user_id serial, username text, password text, i_id int);

WITH x AS (
    INSERT INTO staff(username, password, i_id) -- provide column list
    VALUES ('$username', '$password', nextval('institutions_i_id_seq'))
    RETURNING user_id, i_id
    )
INSERT INTO institutions (i_id, u_id, name)
SELECT x.i_id, x.user_id, '$institution'
FROM   x
RETURNING u_id, i_id; -- if you need the values back, else you are done

staff (u_id serial PRIMARY KEY, ...)
institution (i_id serial PRIMARY KEY, ...)
institution_staff (i_id, u_id, ...,  PRIMARY KEY(i_id, u_id)) -- implements n:m