このたびはご愁傷さまでございます。
Kono tabi wa goshuushousama de gozaimasu.
Please accept my condolences on this sad occasion.

Often heard at both wakes and funerals, this short expression is the one most commonly used for extending one’s sympathy to the bereaved. Because it is so widely used, some people consider it trite, too formulaic, or simply inadequate to effectively communicate a sense of sincere condolence. Nevertheless, in certain situations—such as when signing in at the reception table at a wake—you may find these words extremely useful.

A word about delivery: while it is generally regarded as polite to speak such formal words distinctly and at a deliberate pace, mourn-ers customarily begin this expression by clearly articulating the words kono tabi (translated here as “on this sad occasion”) and then deliver the rest of the sentence (goshuushousama de gozaimasu, or “please accept my condolences”) in hushed, practically inaudible tones deemed appropriate to the unspeakably painful event being observed.

Depending on the circumstances—when addressing a close relative of the deceased, for example—it may be appropriate to add a word or two, such as:

さぞお力落としのことと存じます。
Sazo ochikara-otoshi no koto to zonjimasu.
I’m sure this has been a terrible blow.

心からお悔やみ申しあげます。
Kokoro kara okuyami moshiagemasu.
Allow me to offer my heartfelt sympathy.

These or similar words having been said, decorum dictates that you face the bereaved directly and execute a deep bow from the waist (about 60 degrees) with arms held straight at your sides and eyes forward—don’t bend your neck! Next, raise your eyes slightly without lifting your head. Then lift your head (but don’t stand up straight) to complete your first bow. Finally, bow deeply a second time.

At a funeral, you are likely to be invited to light an incense stick and place it in an urn in front of a photo of the deceased, which is displayed on an altar. You may be invited to do this with an expression such as:

ご霊前にお供えください。
Goreizen ni osonae kudasai.
Please offer a prayer for the soul of the deceased.

This brief ritual has its own explicit protocol. When your turn comes, take an incense stick and light it with one of the matches provided for the purpose, but don’t blow the match out! Gently wave it back and forth until the flame goes out. Place the incense stick, lighted end up, in the urn and bring your hands together in front of your face, head down, in an attitude of prayer. After a moment, bow deeply, then turn and leave the altar.

Before you leave either a wake or a funeral, a member of the bereaved family will probably make a point of thanking you for coming. You are likely to hear:

お忙しい中を、さっそくお悔やみをいただきましてありがとうございます。
Oisogashii naka o, sassoku okuyami o itadakimashite arigatougozaimasu.
Thank you for taking the time, on such short notice, to offer your condolences.

– Source: A handbook of common Japanese phrases –