Air Conditioner Compressor Inoperative
I am suspecting that you do not have the correct power and possibly a capacitor is out as well.

I am going to state a couple observation about your video so read closely.

I am looking at your video post and watching the SLOW moving Condenser Fan and I also heard the BUZZ.

As for the condenser fan it seems like your RUN circuit is not powered up - this can be one of a couple things depending on how that motor is wired - and if it is 220V. Run Capacitor could be bad. However I suspect that you have a BAD phase feeding the AC unit.  The reason being is that your compressor is not running either. 

Please Check your power coming off of your Circuit Breakers - check it at the Terminals of the Condenser Unit . You should have L1 and L2, check L1 - N(115V), L2 -N(115V), L1-G (115V), L2-G (115V) , L1- L2 (230V). I am suspecting a Circuit Breaker has high resistance and either not supplying power or supplying very little power - when measuring at the condenser watch which line (L1 or L2) DROPS when you start the unit up. That will tell you which side of the breaker is bad or possibly your contactor, You will need to measure on both sides of your contactor while doing this check (contactor only pulls in when compressor is told to run) Contactor T1 and Neutral , T2 and Neutral - you will need to verify both sides to know whether you have a bad contactor or a bad circuit breaker. 

If you can post a picture of the wiring diagram found inside the panel cover that will help tremendously. Then I will know how that motor is configured for start winding or not.

When checking your capacitor (always be careful to discharge them before handling as you can get electrocuted [aka Killed] or shocked. To discharge them using a screw driver between Common and Herm or Fan (only handling the insulated part of the screw driver).

If your meter does not check Farads you can test the capacitor in the unit while the unit is "running" - well in your case on. 

Measure the current (amps) of the motor start winding coming off of the capacitor (herm lead and also for the Fan lead) and multiply it times 2652 [in USA 60Hz numbers] then divide that number by the voltage you measure across the capacitor. This will tell the Farads of the capacitor and it should match the capacitor or be with in say 5% , sometimes 10% could be ok depending on the quality of the capacitor. 

With the measurements you made; if you were to assume your capacitor was working you would find what current you should expect across those winding's Herm (yellow according to your diagram) and Fan (Brown according to your diagram).

From your Measurements on the Fan, I used a pseudo value of 1 amp on the winding  using that number your Fan capacitor is at 106 UF but the 5 UF capacitor is a typical proper sizing for a Fan motor. (1 AMP * 2652 ) / 25VAC = 106 UF you can see the amperage I have used is about 21 times too high so 1/21 = .0476 - I am sure that is too low of a current draw to start that motor - so I suspect Capacitor is bad.

Using the same Formula on the HERM (Compressor Side) your measurement was 240V 
So just assuming a low Amperage of 1 AMP (1 AMP * 2652 ) / 240VAC = 11.5 UF your capacitor is supposed to be 50UF or about 4.7 times greater than what our pseudo value gives us. So if you measured with an Amp Meter that the compressor was drawing roughly 4.5 ~ 5.0 Amps on that winding (yellow wire going from Capacitor to compressor) the capacitor for that side would be working.

Given the capacitor is bulging I suspect at least the capacitor has failed especially given your measurements - I know the fan side can't be working if that was truly the fan side you measured. 

Now a suspicion I have about the readings you gave me .. If perhaps you mixed them up and got them backwards.. ~25V was on the HERM side and 240V was on the fan side the fan side would be working and the compressor side would not be working. Bad Capacitor.

50UF should be your compressor side and 5UF should be the AC Fan side. Your part number if my search was correct should be this: PPS550440RD ; However a capacitor that is a Dual 50UF &amp; 5UF rated for 440 VAC will work fine. While your capacitor in the image is 370V , the higher voltage 440VAC has a little more headroom and your situation requires it .. 370V is typical in a 220V circuit here is the formula  1.56 * 220V = 343 VAC, However you have measured 240V which 1.56 * 240 V = 374 VAC and that is OVER the rating of your capacitor and therefore it probably went BULGE no worky after a while. So get one rated at 440CVAC or even 470 VAC.