Get everything in writing
Note down EVERYTHING on your phone call with the recruiters as they may be helpful later on. Even if there are things that are not directly monetary, if they relate to the job, write them down. If they tell you "we're working on porting the front-end to Angular," write that down. If they say they have 20 employees, write that down. You want as much information as you can. You'll forget a lot of this stuff, and it's going to be important in informing your final decision.
Always keep the door open
Never give up your negotiating power until you're absolutely ready to make an informed, deliberate final decision. This means your job is to traverse as many of these decision points as possible without giving up the power to continue negotiating. Very frequently, your interlocutor will try to trick you into making a decision, or tie you to a decision you didn't commit to. You must keep verbally jiu-jitsu-ing out of these antics until you're actually ready to make your final decision.
Information is power
To protect your power in the negotiation, you must protect information as much as possible. A corollary of this rule is that you should not reveal to companies what you're currently making. So given this offer, don't ask for more money or equity or anything of the sort. Don't comment on any specific details of the offer except to clarify them. Companies will ask about your current compensation at different stages in the process—some before they ever interview you, some after they decide to make you an offer. But be mindful of this, and protect information.
"Yeah, [COMPANY_NAME] sounds great! I really thought this was a good fit, and I'm glad that you guys agree. Right now I'm talking with a few other companies so I can't speak to the specific details of the offer until I'm done with the process and get closer to making a decision. But I'm sure we'll be able to find a package that we're both happy with, because I really would love to be a part of the team."
Always be positive
Even if the offer is bad, it's extremely important to remain positive and excited about the company. This is because your excitement is one of your most valuable assets in a negotiation.
Despite whatever is happening in the negotiation, give the company the impression that 1) you still like the company, and that 2) you're still excited to work there, even if the numbers or the money or the timing is not working out. Generally the most convincing thing to signal this is to reiterate you love the mission, the team, or the problem they're working on, and really want to see things work out.
Don't be the decision maker
Even if you don't particularly care what your friends/family/husband/mother thinks, by mentioning them, you're no longer the only person the recruiter needs to win over. There's no point in them trying to bully and intimidate you; the "true decision-maker" is beyond their reach. This is a classic technique in customer support and remediation. It's never the person on the phone's fault, they're just some poor schmuck doing their job. It's not their decision to make. This helps to defuse tension and give them more control of the situation.
I'll look over some of these details and discuss it with my [FAMILY/CLOSE_FRIENDS/SIGNIFICANT_OTHER]. I'll reach out to you if I have any questions. Thanks so much for sharing the good news with me, and I'll be in touch!
It's much harder to pressure someone if they're not the final decision-maker. So take advantage of that.
If you're already in the pipeline with other companies (which you should be if you're doing it right), you should proactively reach out and let them know that you've just received an offer. Try to build a sense of urgency. Regardless of whether you know the expiration date, all offers expire at some point, so take advantage of that.
I just wanted to update you on my own process. I've just received an offer from [COMPANY] which is quite strong. That said, I'm really excited about [YOUR AMAZING COMPANY] and really want to see if we can make it work. Since my timeline is now compressed, is there anything you can do to expedite the process?
Should you specifically mention the company that gave you an offer? Depends. If it's a well-known company or a competitor, then definitely mention it. If it's a no-name or unsexy company, you should just say you received an offer. If it's expiring soon, you should mention that as well.
Either way, send out a letter like this to every single company you're talking to. No matter how hopeless or pointless you think your application is, you want to send this signal to everyone who is considering you in the market.
Companies care that you've received other offers. They care because each company knows that their own process is noisy, and the processes of most other companies are also noisy. But a candidate having multiple offers means that they have multiple weak signals in their favor. Combined, these converge into a much stronger signal than any single interview. It's like knowing that a student has a strong SAT score, and GPA, and won various scholarships. Sure, it's still possible that they're a dunce, but it's much harder for that to be true.
This is not to say that companies respond proportionally to these signals, or that they don't overvalue credentials and brands. They do. But caring about whether you have other offers and valuing you accordingly is completely rational.
Tell other companies that you've received offers. Give them more signals so that they know you're a valued and compelling candidate. And understand why this changes their mind about whether to interview you.
Your goal should be to have as many offers overlapping at the same time as possible. This will maximize your window for negotiating.
Have a strong BATNA (Best Alternative To a Negotiated Agreement) and communicate it.
I 've received another offer from [OTHER CORP] that's very compelling on salary, but I really love the mission of [YOUR COMPANY] and think that it would overall be a better fit for me.
I'm also considering going back to grad school and getting a Master's degree in Postmodern Haberdashery. I'm excited about [YOUR COMPANY] though and would love to join the team, but the package has to make sense if I'm going to forego a life of ironic hatmaking.
Proclaim reasons for everything
It's kind of a brain-hack, both for yourself and for your negotiating partner. Just stating a reason (any reason) makes your request feel human and important. It's not you being greedy, it's you trying to fulfill your goals.
The more unobjectionable and sympathetic your reason, the better. If it's medical expenses, or paying off student loans, or taking care of family, you'll bring tears to their eyes.
Just go with it, state a reason for everything, and you'll find recruiters more willing to become your advocate.
Be motivated by more than just money
You should be motivated by money too of course, but it should be one among many dimensions you're optimizing for. How much training you get, what your first project will be, which team you join, or even who your mentor will be—these are all things you can and should negotiate.
Of course, to negotiate well you need to understand the other side's preferences. You want to make the deal better for both of you.
Understand what they value
Remember that you can always get salary raises as you continue to work at the company, but there's only one point at which you can get a signing bonus.
The easiest thing for a company to give though is stock (if the company offers stock). Companies like giving stock because it invests you in the company and aligns interests. It also shifts some of the risk from the company over to you and burns less cash.
This is more than just giving the company the impression that you like them (which you continually should). But more so that you must give any company you're talking to a clear path on how to win you. Don't bullshit them or play stupid games. Be clear and unequivocal with your preferences and timeline.
Don't waste their time or play games for your own purposes. Even if the company isn't your dream company, you must be able to imagine at least some package they could offer you that would make you sign. If not, politely turn them down.