What is the exact responsibility of the Landing Signal Officer on an aircraft carrier?
An Landing Signal Officer (LSO) is a naval aviator with additional specialized training to better facilitate recovery operations on the ship.  LSOs provide guidance for aircraft making approaches to the carrier.  They monitor the approach and remain in contact with the pilot during the approach by radio.

Carrier approaches or 'passes at the boat', while analogous in technique to an approach to land at a terrestrial airport, require much more precision and have far less margin for error due to the landing area's small size (75 x 600ft). And the requirement that the jet must impact the deck on speed and on angle of attack within a small area to snag an arrester wire and trap successfully makes this even more difficult.

The Navy has adopted this policy of the landing signal officer as well trained LSOs can quickly dissect problems with the approach and alert the pilot to correct prior to the pilot even becoming aware that there is a problem developing.  Particularly insidious errors like settling in the groove or settling at the ramp leading to a dangerous rampstrike or liquid landing can be spotted by the LSO before the pilot even realizes its happening.

In addition to forcing each pilot to perform at their best, the LSO grades each pass.  Multiple LSOs observe the approach and comment on the elements of the approach e.g. Long in the Groove, a little fast, low at the ramp, etc.  These are used to assign the following grades to a pass.


OK UNDERLINE - perfect pass, only awarded in extreme circumstances or
emergencies. Worth 5 points.
OK - excellent approach. On lineup, speed and glidepath with minimal
deviations, each corrected for quickly.  Hook snagged the 3-wire. 
This is the best grade assigned for normal operations. Worth 4 points.
(OK) - A fair pass, exaggerated deviations with delayed corrections,
but approach was safe and conducted properly.  Worth 3 points
OK-BOLTER - an an OK or (OK) pass where the tailhook did not snag a
cross deck pendant.  Worth 2.5 points.
NO GRADE - Approach was safe but had gross deviations and
unacceptable corrections. Worth 2 points
WAVEOFF - Approach terminated, either by the LSO or the pilot for a
variety of unsafe reasons e.g. foul deck, PIO, etc.  worth 1 point.
CUT PASS - Unsafe approach attempted by the pilot, generally after
refusing to comply with a waveoff instruction.  If you survive it,
you can probably expect to have your wings clipped in a Field Naval
Aviator Evaluation Board.  Worth 0 points.


Grades do matter.  A single bad landing or a bolter will get you a razzing by your squadron mates; habitual bad landings and low grades will get you a FNAEB and most likely reassignment to flying helos or to shore based aircraft.  

At least four LSOs will be on hand to observe and grade landings.  This 'Wave Team' will consist of a Controlling LSO, who monitors glideslope and speed, ultimately assigning the grade, a Backup LSO, generally more senior than the Controlling LSO who monitors lineup and assists with grading, and a Deck Status LSO, who monitors the condition of the deck in order to provide a waveoff alert to both the Controlling and Backup LSOs.  A fourth LSO, the Airwing LSO 'Paddles CAG' will be on hand to observe and supervise the whole operation and mentoring junior level LSOs.  There are two Paddles CAGs on every cruise and at least one is on the platform for every recovery.

In the photo above you have a group of LSOs observing approaches during a recovery.  The wired wands that two of them hold are controllers for the waveoff and cut lights on the IFLOLS optical landing aid system to the left of the landing area.  These give visual cues of mandatory instructions from the LSO for a waveoff or for the pilot to cut their engines on the approach.  Each LSO is noting where the aircraft was during the approach vs where it should have been to get a consensus as to what grade and comments to assign to an approach.