Non-Jews are always welcome to attend services in a synagogue, so long as they behave as proper guests. Proselytizing and "witnessing" to the congregation are not proper guest behavior. Would you walk into a stranger's house and criticize the decor? But we always welcome non-Jews who come to synagogue out of genuine curiosity, interest in the service or simply to join a friend in celebration of a Jewish event.
When going to a synagogue, you should dress as you would for church: nicely, formally, and modestly. A man should wear a yarmulke (skullcap) if Jewish men in the congregation do so; yarmulkes are available at the entrance for those who do not have one. In some synagogues, married women should also wear a head covering. A piece of lace sometimes called a "chapel hat" is generally provided for this purpose in synagogues where this is required. Non-Jews should not, however, wear a tallit (prayer shawl) or tefillin, because these items are signs of our obligation to observe Jewish law.
If you are in an Orthodox synagogue, be careful to sit in the right section: men and women are seated separately in an Orthodox synagogue. See The Role of Women in the Synagogue for details.
During services, non-Jews can follow along with the English, which is normally printed side-by-side with the Hebrew in the prayerbook. You may join in with as much or as little of the prayer service as you feel comfortable participating in. You may wish to review Jewish Liturgy before attending the service, to gain a better understanding of what is going on.
Non-Jews should stand whenever the Ark is open and when the Torah is carried to or from the Ark, as a sign of respect for the Torah and for G-d. At any other time where worshippers stand, non-Jews may stand or sit.
When we speak of The Temple, we speak of the place in Jerusalem that was the center of Jewish worship from the time of Solomon to its destruction by the Romans in 70 C.E. This was the one and only place where sacrifices and certain other religious rituals were performed. It was partially destroyed at the time of the Babylonian Exile and rebuilt. The rebuilt temple was known as the Second Temple. The famous "Wailing Wall" (known to Jews as the Western Wall or in Hebrew, the Kotel) is the remains of the western retaining wall of the hill that the Temple was built on. It is as close to the site of the original Sanctuary as Jews can go today. You can see a live picture of the Kotel and learn about it at KotelCam. The Temple was located on a platform above and behind this wall.
Today, the site of The Temple is occupied by the Dome of the Rock (a Muslim shrine for pilgrims) and the Al-Aqsa Mosque (a Muslim house of prayer). The Dome of the Rock is the gold-domed building that figures prominently in most pictures of Jerusalem.
Traditional Jews believe that The Temple will be rebuilt when the Mashiach (Messiah) comes. They eagerly await that day and pray for it continually.
Modern Jews, on the other hand, reject the idea of rebuilding the Temple and resuming sacrifices. They call their houses of prayer "temples," believing that such houses of worship are the only temples we need, the only temples we will ever have, and are equivalent to the Temple in Jerusalem. This idea is very offensive to some traditional Jews, which is why you should be very careful when using the word Temple to describe a Jewish place of worship.