I currently work in HR and do a lot of staffing work, if someone tells me they have another offer, it only makes a difference if I am planning on making that person an offer and I am ready to commit at that time. If I am planning on making an offer, I will try and work in your time schedule to get you the offer before you have to make the decision, if I am not ready to make an offer at that time, it will not persuade me to do so. Do be prepared to provide the date for which other company needs your response. It sounds like the company has already gone through one round of hiring, meaning they already sent offers to their most desired candidates. If a portion of those decline, they will begin offering to their 2nd choices, you are most likely a 2nd choice (not a bad thing, just doesn't give you a lot of leverage)
If you are using another offer to negotiate pay, be prepared to provide the company with that offer. I will not enter into pay negotiations with anyone unless I have proof of the offer, and even then it depends. Companies calcualte pay based on a number of things, the job itself, location, skills the candidate is bringing to the job, some companies pay less but offer better benefits and flexibility. If the counter offer is not for a comparable job in the industry it probably won't get you much, and if you do not provide an offer at all, it won't be taken seriously.