Rickyfox's answer is great in explaining how the weights influence the results of a classifier, but maybe could you be also interested in why / how we would need such weights in the first place (which is more a statistical problem than a purely ML one).
Sometimes the observed data is observed with different distributions and we need to use sampling weights to account for it. You can look at Solon et. al (2015) for more details on why sampling weights matter for analyses and ML (uses mostly algorithms form econometrics literature, but the logic stays the same).
The idea is that these differences in distributions creates imbalances in classes and features. If untreated, this can affect the performance of the predictors / classifiers. I recently wrote a blog post about how you can use these weights to improve the accuracy of some algorithms (features an example with soccer data): https://nc233.com/2018/07/weighting-tricks-for-machine-learning-with-icarus-part-1/
The folowing image shows an example of feature imbalances: these teams of the dataset have not faced the same quality of opposition (elo). The prediction of the rarer types of matchups can be improved by reweighting techniques.
Another example of good use of sampling weights is the treatment of class imbalances (typically when one of the classes is very rare). See for example what is done by default in scikit-learn: http://scikit-learn.org/stable/modules/generated/sklearn.utils.class_weight.compute_sample_weight.html
Finally, despite all these statistical reasons, sometimes we just need to "manually" increase the importance of an observation for very good reasons, and we use the weights to do so :)
Solon, Gary, Steven J. Haider, and Jeffrey M. Wooldridge. "What are we weighting for?." Journal of Human resources 50.2 (2015): 301-316.