There are two ways to pronounce the letter combination “th.”

Both sounds are formed in the same way:

Put your tongue between your teeth and breathe out. 

The difference between the two sounds is this:

1 In words like mother, they or this, the th-sound uses your vocal cords. In your dictionary this sound is written ð.

2 In words like birth, think or thanks, the th-sound doesn’t use your vocal cords. (use only your breath). In your dictionary this sound is written Ɵ.

Have a look at this excellent video:

You should practise this yourself in front of a mirror.

There aren’t any rules to predict which of the two sounds you should use. Sorry!

As far as I know, there is only one word that can be said with either ð or Ɵ, and that is “with.” British people say “with” with ð, whereas Americans say “with” with Ɵ. All the other words with th can be said in only one way.

Here is an exercise:

Try to say the following paragraph:
Where should you say ð and where Ɵ?
Thelma has been thinking her essay through and through, and she’s decided that it is going to argue that thieves are always thin. Her father, Theo, who is a thorough person, takes up a thick pencil. He finds three mistakes in Thelma’s theories. “This essay sucks,” he says. “I think you should throw away your thesis and think of another.” Suddenly there is thunder clapping in the clouds. Thelma and her father seek shelter underneath a tree that is standing tall in their garden. Bathing in the flashing light is a thief who was thin enough to hide behind the tree. “You see?” says Thelma. “Notwithstanding, thieves really are thin.” “No,” thinks her father. “they aren’t always. This is not a rule of thumb.”

Solutions to this exercise: scroll down.

Answers: 

Thelma [Ɵ] has been thinking [Ɵ] her essay through [Ɵ] and through[Ɵ], and she’s decided that [ð] it is going to argue that [ð] thieves [Ɵ] are always thin[Ɵ]. Her father [ð], Theo[Ɵ], who is a thorough [Ɵ] person, takes up a thick [Ɵ] pencil. He finds three [Ɵ] mistakes in Thelma’s [Ɵ] theories [Ɵ]. “This [ð] essay sucks,” he says. “I think [Ɵ] you should throw [Ɵ] away your thesis [Ɵ] and think [Ɵ] of another [ð].” Suddenly there [ð] is thunder [Ɵ] clapping in the [ð] clouds. Thelma [Ɵ] and her father [ð] seek shelter underneath [Ɵ] a tree that [ð] is standing tall in their [ð] garden. Bathing [ð] in the [ð] flashing light is a thief [Ɵ] who was thin [Ɵ] enough to hide behind the [ð] tree. “You see?” says Thelma[Ɵ]. “Notwithstanding [ð], thieves [Ɵ] really are thin [Ɵ].” “No,” thinks [Ɵ] her father [ð]. “they [ð] aren’t always. This [ð] is not a rule of thumb [Ɵ].”