Customs of the Country
Dubai is a fairly liberal emirate, both compared to neighboring parts of the UAE, and to surrounding countries. Many things such as alcohol, dancing, loud music, and scanty clothing are permitted to an extent, but it's best to keep things moderate. Alcohol is sold in bars and restaurants all over the city, but disorderly drunkenness will not be tolerated. There are nightclubs and concerts where people may dance and hook up, but excessive PDAs are a no-no, and you'll get locked up for having a quickie in a public space. A conservative attitude toward nudity and revealing attire prevails. It's fine to walk around the mall in shorts, and bikinis are permitted on the beach, but don't take it too far (no thongs or topless sunbathing), no tank tops for a visit to the mosque. Note that the more remote the area, the more conservative it is likely to be.
Emiratis are in general a warm and generous group of people, but the population in Dubai is filled with international expats. Areas that have a more concentrated local population also tend to be more conservative, and a tourist may not be welcome. It's not strictly polite to approach a person (particularly a woman) in traditional Emirati attire and strike up a conversation. Some may welcome it, but there are still some more private souls who need to be introduced to someone formally before talking to them. Touch is a sign of familiarity, so if you don't know somebody well, it's best to keep your hands to yourself. Children seem to be able to break through this barrier of etiquette, but if you don't have any handy and you want to meet an Emirati, it's best to visit the SMCCU (Sheikh Mohammed Centre for Cultural Understanding) for a cultural breakfast or lunch. You get a meeting, a feed, and a chat all at once.
Around 80% of Dubai's population are expats, and so you'll find that customs and traditions often cross cultures. Half the population are South Asian, so it's a fabulous place to embrace some Subcontinental culture. Trade across to India has thrived for many years, and so you will find that many of the markets, restaurants, and business districts resemble those you would find there. Although everyone gets along in Dubai, you may notice that cultures tend to stick to themselves and have their place in society. One's race can dictate the field of work, salary level, and managerial level one can raise to. For some visitors this is hard to see.
Homosexuality is forbidden in the UAE. This may come as a surprise, because there are plenty of gay celebrities who are known to have visited the country. But you're not going to be refused entry over it, and the fact remains, that you definitely still have the potential for a great time in Dubai, no matter what your sexual preferences are. As with all kinds of PDAs, keep them for later. It's also probably best not to chat up a same-sex customs officer, or start a gay pride march in the street. Interestingly enough, you may see men strolling down the street holding hands, or kissing each other (on the cheeks) warmly in greeting. This is only a sign of friendship for some cultures, much like a slap on the back.
There are rumors that Jews are not allowed into Dubai. The reality is that there is quite a bit of religious freedom in Dubai, provided there is no attempt at conversion, and there are many Jewish visitors and residents (although synagogues are rare). The UAE, however, does not recognize Israel as a country; therefore, if this is your only passport you will not be permitted a visa, except via a personal appeal to the government. If you are intending on visiting Israel before entering the UAE, it is handy to avoid the Israeli stamp in your passport (not all entry points do stamp), however, officially this should not be a barrier.