A consonant interval or chord is one which sounds stable and pleasant. It could, for example, be the end of a piece of music. For example: C and E sound well together, or the chord C, E and G (a C major chord).
A dissonant interval or chord is one which sounds unstable. It may even sound harsh by itself. The notes seem to clash. It cannot be the end of the piece of music (if the music is a normal, traditional tonal piece). It sounds as if it wants to move on to a consonant interval (this is called "resolving" onto a consonant chord). The notes C and Fsharp together make a dissonant interval. The chord C - Dsharp - Fsharp is a dissonant interval: it is full of tension, and it sounds as if it wants to "resolve" onto C - E - G.
Scientifically a consonant interval is one in which the two notes vibrate at frequencies which have a simple mathematical relationship, e.g. in two notes which are an octave apart the higher note vibrates exactly twice as fast. If they are a fifth apart the top note vibrates 1 1/2 times as fast as the lower note. The more complicated mathematically the interval, the more dissonant it will sound.