Unfortunately for the Zerg, both Terran and Protoss players have an easy time countering their units. In the first table in this Starcraft 2 Zerg Counters article, I will discuss what units Terran, Zerg, and Protoss players can build to counter Zerg units.
In the first chart, I have listed all the Zerg units on the left hand side and then the Starcraft 2 Zerg counters in the rest of the chart.
Players looking to defeat the Zerg units on the left hand side should build the other units listed in the same row.
The chart below is organized such that the left side lists Zerg units and the right 3 columns identify which units
easily counter the left side.For example, in the "Roach" row, the units listed in the same row are all
very effective versus Roaches. So, in the "Terran Counter" column next to "Roach", you will see that the Marauder and Siege Tank are both listed. The Siege Tank and Marauder are very strong versus the Roach, hence they counter it.
To see which Zerg units beat other races' units, please see the second chart.

In the second Starcraft 2 Zerg counters chart (below) highlights which units Zerg players should build in order to counter the enemy. So, if the left heading says "Zergling", you will want to build Zerglings to defeat all other units which are listed in the same row.

These two Starcraft 2 Zerg counters tables should help you out significantly in your online play. The strategy is simple, if you play as Zerg, use the first chart so you can figure out what to avoid building.
If you are playing as Zerg, you can get the idea of what Zerg units are strong against other enemy unit types. After scouting, you can change to whatever build you want based off your enemy's actions.
At first, this might take a long time to learn. After all, there are quite a few unit counters in game. Just keep playing and practicing the creation of units which counter the enemy. If you are a Zerg player or struggle beating Zerg players, I would recommend printing these charts out.
Then, as you are playing, you could reference the tables to see what you are supposed to build (or avoid building) based on what units the enemy has.
These charts are more important than any other race than Zerg. Since Zerg can rapidly go between producing dozens of air units to dozens of ground units, being familiar with all the Starcraft 2 Zerg counters can help you easily switch between unit types based on what your enemy has and what he is planning to get.