For example filters in signal/image processing can be mathematically described by convolution. The inverse operation (deconvolution) is used e.g. in some kinds of microscopy.

Correlation... well... everywhere? Basically whenever you want to quantify how one thing affects another...

correlation is used to extract second (and higher) order statistics from any random signal, (de)convolution is inherent in any operation where the Fourier transform is taken with irregular sampling: one real life example = interferometry

In my research (analyzing network-based intrusion alerts), correlation is the only promising method to find significant relationships among alerts that have been triggered by multiple intrusion detection sensors. Security Admin (SA) needs to understand and study these alerts. They are meaningless if being analyzed individually. Somehow, they must be 'connected' with previous alerts or future alerts. So SA can figure out the sequences of attacks that have been launched on the network. This is important to identify preventive measure in the future.

The basic application of the convolution is to determine the response y[n] of a system of a known impulse response h[n] for a given input signal x[n]. to obtain y[n] you just have to calculate the convolution of the x[n] and h[n].

One of the real life applications of convolution is seismic signals for oil exploration. Here a perturbation is produced in the surface of the area to be analized. The signal travel underground producing reflexions at each layer. This reflexions are measured in the surface through a sensors network. The signal obtained is  the convolution of the reflexion coefficients times the the perturbation. 

Convolution is a basic operation of linear systems. Given a linear system H and an input X, the output is Y = H ⭐︎ X, where ⭐︎ denotes convolution. Convolution is ubiquitous in linear systems.

check the following link:

http://www.wavemetrics.com/products/igorpro/dataanalysis/signalprocessing/convolution.htm

Multiplication of big integers is another application

http://en.wikipedia.org/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm

Any scientific field that requires mathematical computation of data in form of 'multiply and accumulate' and 'accumulate and multiply' can be done using Convolution and correlation. They are frequently used in communication systems and forms the theoretical basis of Digital Signal processing .

The convolution of two probability distribution functions gives the pdf of the sum of the two random variables. The cross-correlation of the two pdfs gives the pdf of the subtraction of one random variable from the other. This "explains" why convolution is commutative and cross-correlation is not.

You could visualize the two operations like this ...

Convolution is what you would use to obtain a new position distribution from an old velocity distribution and an old position distribution.

Correlation is how you would obtain a new velocity distribution from a new position distribution and an old position distribution. 

deterministic signals:

- the convolution is used to compute the response of a specified linear time invariant system to a specified signal,

- the correlation is used to appreciate the similarity between a signal and its traslated variants,

random signals:

- the convolution is used to compute the probability density function of a sum of two independent random variables (based on their probability density functions),

- the correlation is used to appreciate the stationarity of a random signal.

A main application field of these tools can be found in communications, for instance in the digital receivers. The radar systems are another field where correlation is the vehicle to map the distances

Real life examples where the convolution is important are:

- observed optical transmission or absorption spectrum is a convolution of the true signal with detector's spectral window

- the slowly varying voltage measured by usual equipment is a convolution of a true signal with "measuring time window", which has in this case the shape ~exp((t-t_0)/\tau)) for t < t_0 (t_0 is the moment of a measurement) and zero otherwise, and \tau is time constant of the measuring instrument.

Needless to say, the ever present convolutions always distort measurements.

Correlation is applicable whenever you are interested in finding statistical relationships between quantitative variebles and later predict one of them from another.

Then, you have many different applications in biology, social sciences or education, for example

 check out this video, very simple example on convolution

https://www.youtube.com/watch?v=8jCva6KHlYI 

The relation between the employee experience & its productivity. 

If you want to use a slit to measure the varying brightness along a line of light, then the result is the convolution of the variation you want to measure with the rectangular function describing the sampling by the slit.  Convolution here is a practical difficulty, and de-convolution would be helpful.  The same goes for using a radar beam to measure what is out there - the result is what you are really interested in convolved with the shape of the radar beam.

Correlation of pseudorandom binary codes is what makes GPS work, lots of radar systems, and lots CDMA (code division multiple access) systems.  It is why GPS is power hungry, it takes a lot of processing to find the correct correlation, with correct satellite (correct code) and the correct delay.

Filtering, modulation for analog and digital signals , demodulation for both analog and digital signals, image processing.

in addition to Amani answer: blind signal processing and target detection

in radar