Most representative democracies come in one of two types: A presidential system such as the USA, in which there is an executive who is directly elected, and a parliamentary system such as the UK, in which the legislature is elected and then the legislature chooses the executive. Some countries have hybrid systems (such as France), and other systems are possible (the executive could be folded fully into the legislature, made a council of several people, etc.), but these are by far the most common.
A presidential system like the US is inherently more democratic; any time you can pass the power directly to the people instead of making it through some intermediary you make the system more democratic. But "more democratic" does not necessarily mean "better"; many political scientists argue that it is possible to have too much democracy, and end up electing demagogues and enacting awful policies by rallying the opinion of an ignorant public.
A parliamentary system can be a means of weakening the tyranny of the majority, the possibility that a democratic vote could lead a majority of the population to violate the rights of a minority of the population. With a parliamentary system, there is more coalition-building within the parliament necessary to achieve executive power, mitigating the effects of simply having a majority in favor of a harmful policy.
Parliamentary systems generally avoid partisan gridlock, where one party controls the executive and an opposing party controls the legislature, and neither is willing to do anything because it would mean supporting the opposing agenda. (This has clearly been a major issue lately in the US, whereas it rarely happens in the UK.) On the other hand, parliamentary systems remove a check and balance---the executive and legislative branches are now more closely tied, and the two cannot as easily act as checks on one another as they could in a presidential system. So again it depends on whether we would rather have government that does things the people might not agree with, or one that doesn't do anything at all.
There is also empirical evidence suggesting that parliamentary systems may be more stable against coups and transitions to authoritarianism, particularly in countries with high levels of ethnic, religious, or cultural fractionalization. Third World countries that used parliamentary systems have had much more success transitioning to democracy than those that used presidential systems. This could be for many reasons, but all other things equal, it does argue in favor of a parliamentary system.
Some parliamentary systems allow the parliament to schedule elections more or less when they please (within certain constraints), whereas presidential systems almost always have a well-defined election schedule such as holding an election every four years. This can allow the ruling party to partially control the election outcome by manipulating the calendar, holding votes only when their poll numbers are favorable. On the other hand, it avoids the inconvenience of overly frequent elections and allows new elections to be held if a prime minister is incapacitated (whereas in a presidential system there usually has to be a vice president or some similar mechanism for automatic succession).