Hi
DCT is the type of Fourier transform, in this transform cosines are basis functions,where as in wavelet transform wavelet is the basis funtion.that means in DCT your signal is represented in terms of cos functions where as in wavelet transform signal is represented with shifted and scled version of mother wavelet.
Beauty of wavelet transform is time -frequency representation and basis choice.
Added after 2 minutes:
Hi
Sorry i forgot to mention about applications of DCT and wavelet transform(WT)
mostly DCT is used in Image compression, where as wavelet transform is also useful in image compression, Image denoising,signal denoising,Transient signal processing,time-freq representation.