Are mutations random?
Short answer
The claim is unclear but is essentially misleading and wrong. However, IMO, for lay people, it is a good approximation to just think that mutations are random!
Here, on Understanding evolution is a great source of information on what it means to say that mutations are random
Long answer
What is a mutation?
Mutation is an event causing an alteration in the DNA sequence where the mutation occurred. There are different types of mutations, consider for example the following Single Nucleotide Polymorphism (SNP)
Original Sequence

ATTATCTG

Sequence after mutation of the 3rd nucleotide

ATGATCTG

Note that mutations can have much more drastic effects such as the duplication of a very long sequence of DNA for example.
You might want to have a look at What is the definition of a mutation? and eventually Effect of mutation on phenotype.
What can be random?
A variable can be random. So is the term mutation a variable?
IS &quot;MUTATION&quot; A VARIABLE
Consider first this discussion about the term house. You can measure the height of the house or the number of windows and both the height of the house and the number of windows so that both the number of windows and the house of houses are variables but house itself is not a variable.
A mutation is an event but, just like with the house, there is no obvious number associated with the term mutations. The number of mutations occurring in a reproductive event is a variable. The effect size on fitness of a mutation is a variable. So claiming that mutations are random is unclear as we don't know what property of the mutation the claim is referring to.
WHAT IS RANDOM
Fully discussing the concept of randomness will require doing quite a bit of math and philosophy. I will not attempt to do so and will intentionally make some oversimplifications here.
The concept of randomness only makes sense if we formalize a model. The outcome of flipping a coing may seem random to you but this is true only if you assume you don't know the forces at play on the coin (the movement of the thumb, the speed of the wind, etc...). A physicist would be happy to consider a coin flip being a perfectly deterministic (deterministic = not random) process.
If a process is deterministic, it means that for a given input there is a single possible output. If a process is stochastic, then there are several (potentially an infinite number) of possible outputs. Now, stochastic does not mean fair or uniformly distributed and therefore a process can be stochastic and yet we can have some predictive power on what range of values the outcome will likely take. For example a weighted die/dice may turn up certain numbers more often, but it's still random, it's just that the distribution of expected outcomes is not uniform.
So, to talk about the claim mutations are random, ideally one would need to fully specify a model of consideration.
RANDOM AS DIRECTIONLESS
Often by random, people mean &quot;directionless&quot; (or &quot;fair&quot;). This is clearly not what the term random means but I will still discuss the case of &quot;direction&quot; (or &quot;fairness&quot;) as it is often what people mean. If a mutation is directionless, it means that this mutation is not more likely to increase the value of the variable than to decrease it. For example, if the variable of interest is the height of the individual, assuming that mutations are truly directionless, a mutation is not more likely to increase the height of an individual than to reduce it.
Is the number of mutations random?
VARIATION IN MUTATION RATE
The mutation rate varies dramatically among species and along genomes roughly speaking from a rate of $10^{−2}$ to a rate of $10^{-10}$ mutation per nucleotide. For example microsatellites (repeated sequences) are highly mutable.
EFFECTS OF NEARBY SITES
Note also that the probability of a given nucleotide to mutate depends also on the exact nucleotides that are around it (Rob Ness; Personal communication).
PLASTICITY FOR MUTATION RATE
The mutation rate is also a plastic trait (and is therefore under selection) in some species (of plants and bacteria typically) (Chary et al. 1993; Bjedov et al. 2003). The environment affects the mutation rate and the environment can be chosen by some species (avoidance) or even constructed by others (niche construction).
AGE OF THE PARENTS
In multicellular organisms, the age of the father correlates with the number of new transmitted mutations as well. See the post Gender and age-specific mutation rate in plants (and its references) for more information.
EPIGENETICS
Epigenetic changes (do not concern all species) can also affect the mutation rate and even influence what new nucleotide the old nucleotide will be mutated into (with a given probability).
CONCLUSION
Many things affect the mutation rate. However, the exact number of new mutations an offspring will carry is not perfectly deterministic.
Type of mutations
Mutations are more likely to be transitions than transversions. But again, it is not a perfectly deterministic process.
Effect of mutations on fitness
DISTRIBUTION OF FITNESS EFFECTS
Different mutations have different effects on fitness. We talk about the Distribution of Fitness Effects (DFE).
The DFE varies from one sequence to another. In conserved sequences, mutations tend to have a higher impact on fitness.
ARE MUTATION EFFECTS ON THE FITNESS DIRECTIONLESS?
No, most mutations are deleterious or neutral. Only few mutations are beneficial.
Effect of mutations on the phenotype
Again, some phenotypes are more sensitive to new mutations than others. The increase in phenotypic variance due to mutation at every generation is called the mutational variance.
ARE MUTATION EFFECTS ON THE PHENOTYPE DIRECTIONLESS?
It can be the case, but it is not necessarily true. When mutations are not directionless, we talk about mutation bias (see for example Eyre-Walker 1998, Harr and Schloterrer 2000 or Shah et al. 2010).
Conclusion
Yes, there is some randomness in many of the variables associated with mutation events. However, it would be wrong to think that we have no predictive abilities in the number of mutations or effect of mutations, it would be wrong to think that a mutation is as likely to increase fitness than decrease it and it could be wrong to think that a mutation is as likely to increase a phenotypic trait than decreasing it.
Related posts

Are evolutionary mutations spontaneous?
What are the major causes of mutations in DNA?

Source of information in evolutionary biology
Understanding Evolution by Uc Berkeley is a good, short and very introductory course to evolutionary biology.