According to this talk by Elder Ballard, many non-LDS confuse LDS temples, tabernacles, and churches. They know that non-LDS aren’t allowed to enter, uh, churches? Wait, temples? I know this…
Lots of people are confused. So, I will clarify.
Local congregations or “wards” (you can find out about what meetings are like and locate the nearest ward to you here) of the LDS Church meet weekly in churches, often referred to as “the church,” like ” Hey, I’ll see you at the church for the meeting.” Anyone can attend, visit, walk in, etc. Smaller units are called “branches” but still meet in a church, if they have one. Otherwise, they may meet in someone’s house or a rented building.
A collection of wards branches is known as a Stake (drawn from Isaiah and the book of Doctrine and Covenants). One of the church buildings in the geographical stake is designed to accommodate larger meetings of multiple wards and branches at the same time, at a twice-yearly meeting called Stake Conference. This larger building (which most often resembles a normal church except in size) is known as a Stake Center. There are church buildings and stake centers throughout the world.
Tabernacles are rare, and haven’t been built for a long time. In the Bible, the tabernacle was the temporary and portable temple, but also the “tent of meeting” for the Israelites. Tabernacles in early LDS history served the same purposes as Stake Centers today, in allowing larger numbers of LDS to congregate together. The Salt Lake Tabernacle used to be where the world-wide twice-yearly General Conference was held and broadcast from, and the Mormon Tabernacle Choir sang. (A new building has recently replaced the Tabernacle to accommodate larger numbers of people, the Conference Center.) Tabernacles are open to anyone.
Temples, by contrast, are not used for regular weekly meetings, but are reserved for special ordinances for active LDS who meet certain standards of belief, practice, and ethics. Prior to the dedication of a temple, anyone can tour it, but after the dedication of the Temple, it is reserved for those who meet the standards we believe God has established. Pictures of some interiors have been published. (See here for pictures of the Salt Lake Temple interior, for example.)
So, to sum up- Churches and tabernacles are open to anyone, and there are far more of the former than the latter. Temples are open to anyone prior to being dedicated, but are afterwards limited to LDS who live the standards we believe God sets in order to enter his house.