The golden rule with all complaints is to stay calm. Never take a complaint personally, even when it feels like that's how it was meant. Remember that a client may be desperately worried about their pet, and unable to deal with this emotionally. Some will 'take it out' on veterinary personnel simply because they feel that they need someone to blame. Whatever the rights and wrongs, it is your job to rise above it.
The easiest and quickest way to defuse many client complaints is to apologise unreservedly, regardless of where blame lies. The client is, after all, always right! These days, though, litigation has become increasingly commonplace, and an apology for a particular situation may be taken as an admission of liability.
It is therefore important to exercise good judgement as to when and how an apology is made. Regardless of the situation, you can and should always apologise for the fact that the client is unhappy. This implies no admission that you or the practice are in any way at fault.
However, you should only apologise for the problem itself if it is a minor one. If the problem is a serious (say if it relates to the wellbeing of a patient), it is for senior staff or partners to decide whether an apology is appropriate, and to deliver it, if it is.
No two clients are ever the same, so you will probably find that there are different ways of dealing with identical complaints. Here are some broad pointers: