How distances are measured in the Solar System, in the Galaxy, and in the Universe? Distances in our universe can be calculated in several ways: 


Stellar Parallax
Standard Candles
Redshift


How are these methods linked, and how they are actually used to calculate   distances are measured in the Solar System, in the Galaxy, and in the Universe?
Stellar Parallax
Stellar parallax uses differences in perspective to determine the distance from an object. When the earth goes around the sun, our perspective of the star, galaxy etc. changes and so the angle from us to the object changes. Because we know how the earth moves around the sun, we know the distance between the points that we take the measurements. From this we can use the Sine Rule to determine the distance from the object.
For Example, if the distance between the two points of measurement were 2 units, and the two inside angles (A and B) were (A)80° and (B)75° (and therefore the other angle (C) is 25°) we can say that:
$a = \frac{2\sin80}{\sin25}\ $
where &quot;a&quot; is the opposite side to angle &quot;A&quot;. This means that the distance between the object and the second (B) point of measurement is approximately 4.66 units.
Diagram here.

Standard Candles
 A standard candle is an astronomical object that has a known absolute magnitude. (source)  
By using the known absolute and the measured apparent magnitude (what is seen by instruments etc.) it is possible to determine the distance of the object by using the formula:

m-M = 5 log d – 5
where m is the apparent magnitude of the object, M is the absolute magnitude of the object, and d is the distance to the object in parsecs.(same source)

One of the most widely used examples of these standard candles is the supernovae type Ia, which was used to discover that the universe is accelerating in its expansion. This led to the awarding of the 2011 Nobel Prize in Physics to Saul Perlmutter, Brian Schmidt , and Adam Riess.
Additionally, RR Lyrae are another form of standard candle used to measure galactic distances (SN Ia are much more luminous and therefore can be used to much higher distances). RR Lyrae are a type of variable star with a relationship between pulsation period and absolute magnitude, which is what makes them good for use as standard candles.
Other information here.

Redshift
Redshift is the effect where light from an object is &quot;stretched&quot; out so that the wavelength is moved towards the &quot;red&quot; end of the spectrum. Distances are calculated by using Hubble's Law, but I'll let others explain it: here, here and here.