Using Oracle SQL, how does one output day number of week and day of week?

select level as dow, 
    to_char(level+trunc(sysdate,'D'),'Day') as day
from dual
connect by level <= 7;

Florin's answer is how I'd do it, but you need to be a little careful about NLS settings. The day of the week is affected by the NLS territory, so if I run this as if I'm in the US it works:

alter session set nls_territory = 'AMERICA';

select to_char(sysdate, 'D') as d, to_char(sysdate, 'Day') as day from dual;

D DAY
- ------------------------------------
6 Friday

select level as dow,
    to_char(trunc(sysdate ,'D') + level, 'Day') as day
from dual
connect by level <= 7;

DOW DAY
--- ------------------------------------
  1 Monday
  2 Tuesday
  3 Wednesday
  4 Thursday
  5 Friday
  6 Saturday
  7 Sunday

But the same query run in the UK is a day off:

alter session set nls_territory = 'UNITED KINGDOM';

select to_char(sysdate, 'D') as d, to_char(sysdate, 'Day') as day from dual;

D DAY
- ------------------------------------
5 Friday

select level as dow,
    to_char(trunc(sysdate ,'D') + level, 'Day') as day
from dual
connect by level <= 7;

DOW DAY
--- ------------------------------------
  1 Tuesday
  2 Wednesday
  3 Thursday
  4 Friday
  5 Saturday
  6 Sunday
  7 Monday

... and I need to adjust the calculation to correct for that:

select level as dow,
    to_char(trunc(sysdate ,'D') + level - 1, 'Day') as day
from dual
connect by level <= 7;

DOW DAY
--- ------------------------------------
  1 Monday
  2 Tuesday
  3 Wednesday
  4 Thursday
  5 Friday
  6 Saturday
  7 Sunday

You can also specify the language used for the day names separately if you want:

select level as dow,
    to_char(trunc(sysdate ,'day') + level - 1, 'Day',
        'NLS_DATE_LANGUAGE=FRENCH') as day
from dual
connect by level <= 7;

DOW DAY
--- --------------------------------
  1 Lundi
  2 Mardi
  3 Mercredi
  4 Jeudi
  5 Vendredi
  6 Samedi
  7 Dimanche

Documentation for to_char() with nls_date_language and day of the week, and more in the globalisation support guide.