Considering that the moon’s gravitational pull on earth is so weak- we need special instruments to detect it- how does it cause the tides? That the moon’s gravitational pull is responsible for the tides is established. However, if we want to detect the moon’s gravitational pull on earth, this is very challenging. This seems to be a paradox considering the enormous force needed to drive the tide.
Gravity is an acceleration, and force = mass x acceleration.  Thus, if we want to detect the Moon's gravitational pull, we need either a giant mass or a really precise instrument.  Since the oceans are a giant mass, the force on the ocean is thus correspondingly large.  The fluidness of the ocean also allows it to move, even if the force per volume is small.
To expand on this, the tidal acceleration due to the Moon on the surface of the Earth is about 1 μm/s$^2$ (1), which means that the force due to the weight of a 1000 kg object will vary by about 0.001 N over the course of a day.  1000 kg of water is about 1 cubic meter, so every cubic meter of the ocean experiences that tiny variation in force as well.  Since the ocean has about 1.3 x 10$^{18}$ cubic meters (2) of water in it, you could say the tidal force is 1.4 x 10$^{15}$ N, which is an absolutely enormous force - translated into weight, it's more than the weight of all living things on the Earth (3).
There is also a corresponding acceleration from the Sun, about half as strong as the Moon, which also has some effect on the tides, resulting in a monthly very high and very low tide, known as the spring tide, and a somewhat averaged tide, known as the neap tide.  It might also help to realize that this 1/2 as strong force is what is also responsible for keeping the whole Earth in orbit around the Sun.