It depends on who the connection is.
Who is the person you want to use as a connection within the firm to which you are applying? Is the person an attorney who is in the group to which you are applying? Is she a partner in that group? An associate? The position of the person will help determine when to use that connection.
If the connection is a partner who may influence whether you get interview and/or hired, you may want to contact that person shortly after your resume has been sent to the firm by your recruiter. Often, recruiting coordinators (who are sometimes non-lawyers) serve as gatekeepers within a law firm, and determine whether your resume gets seen at all by the relevant attorneys. Therefore, it may be in your interest to let that person know soon after your resume has been sent that it has been submitted. That way, you increase your chances of the resume being sent on to the group by the recruiting coordinator. Even if the connection is a partner in another group, who may not be someone who has any say in whether you get an interview or an offer, it may not hurt to let that person know your resume has been sent. She probably knows a partner in the group who has the power to make sure it is seen.
Even an associate can help your cause from the outset, but the key factor is making sure that the associate is well-regarded by the firm. Unlike partners, being an associate does not guarantee that the person is doing well. Partners have been invited to join an elite and private club in a very real way; associates are still proving themselves with the hope of being invited, so you really never know whether a recommendation from an associate will be a help to you or not. As a general rule of thumb, if the associate is fairly senior, or has been with the firm more than 3-4 years, she is probably doing pretty well there, and you have a reasonable chance of a connection helping, rather than hurting you. This is especially true if she is in the group to which you are applying. If she is in another group, there is a smaller chance that she will have a relationship to the partners who will make decisions with regard to your candidacy, but you can always ask. The last thing you want to do is attach yourself to an associate who is on the way out the door.
It depends upon what the connection is.
What is this person’s connection to you? Is the relationship a professional one? A personal one? Is this person someone you knew in law school? Is the person only an acquaintance? The most important factor in all of the above, which I can’t stress enough, is whether you can count on this person saying nice things about you.
The best connection, of course, is someone who has worked with you and can vouch for your work product and work ethic. Again, if you have such a connection at a firm to which you have applied, I highly recommend that you reach out to that person early on in the process to let her know that you have applied to her firm. If you are not 100% sure that she will say wonderful things about you, call her up or ask to take her to coffee, and try to get a feel for how she feels about your candidacy there. Tell her that you are very excited about the opportunity and think you would be a great fit, and ask if she would mind discussing with the firm the projects you worked on with her. Remind her what those projects were, and what your role was. Stress the projects where you distinguished yourself. In any event, you do need to contact that person if she is in the group that will be considering your candidacy, since your prior relationship to her will probably come to light at some point, and they will ask her about you. You want to make sure that you are shaping that conversation by speaking to her first.
Even a purely personal relationship can be very helpful. If someone can say that they have known you for many years on a personal level, and can vouch for your character, as it were, this can go a long way, especially in smaller firms. Many firms place a high value on personality fit. This can be especially helpful at the end of the process.
Many attorneys I have worked with, especially associates, make the mistake of dropping names of acquaintances during an interview without permission. I do not recommend this. If you have an acquaintance or former law school classmate who is at the firm to which you applied, you should shoot that person an email and let them know you will be interviewing there. Do not be the person who avoids contacting your acquaintance because you feel awkward doing it, only to drop that person’s name without her permission. Most people do not appreciate this.
It depends upon where in the interview process you are.
As discussed above, many connections are best used at the beginning of the process, after the resume has been sent by your recruiter. However, an associate in an unrelated group may actually help you more at the end of the process, or after you already have secured an interview. This is because an Associate in an unrelated group amounts essentially to a personal recommendation, since they will probably have little to say about the substance of your work.
In sum, think about the type and closeness of the relationship you have, and be sure to contact your connection before you offer up her name during an interview. Always, always do everything you can to ensure a positive review of you before you mention the connection to anyone.