The perfect fourth spans 5 semitones and has a pitch ratio of 4/3. The perfect fifth spans 7 semitones and has a pitch ratio of 3/2. They are called "perfect" because they occur in both major and minor scales and were always considered as consonances.
We hear the fourth in the opening of a lot of tunes before the root on the strong beat. The anthems are the best example. La Marseillaise starts with a fourth.
On the other hand, a lot of guitar solos start with a fifth. The solo of Stairway to Heaven is a famous example.
The ascending fourth sounds like an answer, while the ascending fifth sounds like a question. The situation is opposite with the descending fourth and fifth. The descending fourth sounds like a question, while the descending fifth sounds like an answer.
It's also easy to hear the fourths and the fifths together as double-stops. The fourth occurs between the fifth and the root of major and minor chords (G - C in C major, for example). The fifth occurs between the root and the fifth of major and minor chords (C - G in C major, for example). They also form the "power chords", so often met in rock.
Between the perfect fourth and the perfect fifth we find the dissonant augmented fourt or diminished fifth, known also as the tritone. It spans 6 semitones and has a pitch ratio of 7/5.
The octave inversion of the perfect fourth is the perfect fifth, while the octave inversion of the perfect fifth is the perfect fourth. The tritone is its own octave inversion because it divides the octave in two equal parts.
It's very easy to hear a tritone. A song that starts with an augmented fourth is Maria from West Side Story. It may be heard in a lot of rock riffs like Metallica's "Enter Sandman", in Hendrix' "Purple Haze", in Dream Theater's "As I Am", etc.
To hear the tritone as double-stop, just think that it occurs between the third and the seventh of the dominant seventh (B - F in the G7 chord, for example).