How to perform SQL queries and get results from QGIS python console?

1) With PyQt4.QtSql: Gary Sherman shows how to query a spatial table in Creating a PostgreSQL Connection From a QGIS Layer Datasource:

from PyQt4.QtSql import *
layer = iface.activeLayer()
uri = QgsDataSourceURI()
uri.setConnection("localhost", "5432", "testpostgis", "me", "")
uri.setDataSource("public", "teststrati", "the_geom")
# add the layer to the canvas
vlayer = QgsVectorLayer(uri.uri(), "tot", "postgres")
# now query the table
db = QSqlDatabase.addDatabase("QPSQL");
db.setHostName(uri.host())
db.setDatabaseName(uri.database())
db.setPort(int(uri.port()))
db.setUserName(uri.username())
db.setPassword(uri.password())
db.open()
# query the table
query = db.exec_("""select * from teststrati""")
query.next()
query.value(0)
130
# etc read the documentation of QtSQL

And you can open all the other tables/views (spatial or non spatial) similarly:

db = QSqlDatabase.addDatabase("QPSQL")
db.setHostName("localhost")
db.setPort(5432)
# non spatial table or view
db.setDatabaseName("people")
db.setUserName("me")
db.setPassword("")
query = QSqlQuery(db)
query.exec_("select * from people;")
# etc.

2) Or you can use the standard Python module for PostgreSQL/PostGIS: Psycopg2:

import psycopg2
conn = psycopg2.connect("dbname='testpostgis'host='localhost' user='me'")
cur = conn.cursor()
sql = """SELECT "DIP_DIR","DIP", ST_AsGeoJSON(the_geom) from teststrati;"""
cur.execute(sql)
result = cur.fetchone()
print result
(130, 30, u'{"type":"Point","coordinates":[272070.600040999997873,155389.387920000008307]}')

With a non spatial table or view and the result as a dictionary:

conn = psycopg2.connect("dbname='testpostgis'host='localhost' user='me'")  
dict_cur = conn.cursor(cursor_factory=psycopg2.extras.DictCursor)
dict_cur.execute("""SELECT * from people;""")
rec = dict_cur.fetchone()
rec.keys()
['name', 'id']
rec.values()
('Jon Doe',1)
# etc read the documentation of the module