Vowels are the backbone of Japanese pronunciation. In order to master Japanese pronunciation you have to master the vowels. Luckily for non-native learners, there are only 5 of them. However, these vowels are quite different in quality to the vowels found in English and will take practice to get perfect.
Standard American English has two types of vowels, monophthongs and diphthongs. Monophthongs are pure vowels; your mouth and tongue do not move much while pronouncing them and they sound like one single sound. An example is the <u> in the word cut. Diphthongs on the other hand are gliding vowels; your lips and tongue continuously move throughout the duration of the vowel resulting in a smooth interpolation between the starting vowel and the ending vowel. An example of this in English is the <o> in boat. If you say this word is slow motion, you will see your lips move all the way from the end of the <b> sound to the beginning of the <t> sound.
Japanese, by contrast, only has monophthongs, meaning all vowels are pure. In fact, I would go so far as to say that they are even more pure than in English, since even English monophthongs allow for a tiny bit of wiggle room in terms of mouth movement. When pronouncing Japanese vowels, your lips and tongue should essentially be locked in place.
Getting this down is crucial, as one of the biggest giveaways of an English accent is sloppy vowels. For example, the Japanese vowel <o> sounds closest to the English <o> in the word boat described above. However, the <o> in Japanese is a strict monophthong, while the English <o> is a diphthong. As a result, there is a tendency amongst English speakers to replace the Japanese <o> with the English <o> because it is easier for them to say. The end result is a pronunciation that may or may not be understood by the person you are talking to.
Now that I have gone over some common pitfalls for English speakers, I will go over each vowels.
The first vowel is あ and is pronounced similar to the <a> in Father.
The second vowel is い and is pronounced similar to the <i> in Beet.
The third vowel is う and is a sound unique to Japanese. It sounds similar to the <u> in boot but with the lips more tightly compressed.
The fourth vowel is え and sounds similar to the <e> in bet.
And lastly, the fifth vowel is お and as mentioned before sounds similar to the <o> in boat but is a pure monophthong instead of a diphthong.