A cover letter requirement varies significantly as per the geography of application. If you are applying through a referral in the United States, then a cover letter becomes mandatory - not necessarily so in other parts as pointed out by my fellow experts. But, since you have such a doubt, it is always better to make that extra effort than not.
I would suggest you prepare the following:
1. A short and crisp cover letter for the particular firm where you are applying
2. A generic cover-letter for future - this can be a generic version of the cover letter you prepare for this firm
Please do keep in mind the following points while drafting your cover letter:
- Be precise and to the point: Do not beat around the bush or try to use flowery language. Understand that consultants are busy folks and they will not be able to read the entire thing if you write long sentences.
- Structure: This is the key-word to consulting. Structure your cover letter into separate paragraphs with each paragraph centred around different ideas. This has been covered greatly by Francesco.
- Give them what they want: Don't just talk about yourself and your past achievements. Highlight what is it in for them and why they should hire you. Have just 2-3 illustrative examples to show that you are the right fit.
- Gratitude: Finally, thank them for giving you this opportunity and for any past interactions they might have had with you.
Wishing you all the best. Please do get in touch if you need any other assistance.