The Jobs Argument
One of the main arguments against free trade is that, when trade introduces lower cost international competitors, it puts domestic producers out of business. While this argument isn't technically incorrect, it is short sighted. When looking at the free trade issue more broadly, on the other hand, it becomes clear that there are two other important considerations. First, the loss of domestic jobs is coupled with reductions in prices of goods that consumers buy, and these benefits shouldn't be ignored when weighing the tradeoffs involved in protecting domestic production versus free trade. Second, free trade not only reduces jobs in some industries, but it also creates jobs in other industries. This dynamic occurs both because there are usually industries where the domestic producers end up being exporters (which increases employment) and because the increased income held by foreigners who benefited from free trade is at least partly used to buy domestic goods, which also increases employment.
The National Security Argument
Another common argument against free trade is that it is risky to depend on potentially hostile countries for vital goods and services. Under this argument, certain industries should be protected in the interests of national security. While this argument is also not technically incorrect, it is often applied much more broadly than it should be in order to preserve the interests of producers and special interests at the expense of consumers.
The Infant-Industry Argument
In some industries, pretty significant learning curves exist such that production efficiency increases rapidly as a company stays in business longer and gets better at what it is doing. In these cases, companies often lobby for temporary protection from international competition so that they can have a chance to catch up and be competitive. Theoretically, these companies should be willing to incur short-term losses if the long-term gains are substantial enough, and thus shouldn't need assistance from the government. In some cases, however, companies are liquidity constrained enough that it can't weather the short-term losses, but, in those cases, it makes more sense for governments to provide liquidity via loans than to provide trade protection.
The Strategic-Protection Argument
Some proponents of trade restrictions argue that the threat of tariffs, quotas, and the like can be used as a bargaining chip in international negotiations. In reality, this is often a risky and unproductive strategy, largely because threatening to take action that is not in a nation's best interest is often viewed as a non-credible threat.
The Unfair-Competition Argument
People often like to point out that it's not fair to allow competition from other nations because other countries don't necessarily play by the same rules, have the same costs of production, and so on. These people are correct in that it's not fair, but what they don't realize is that the lack of fairness actually helps them rather than hurts them. Logically, if another country is taking actions to keep its prices low, domestic consumers benefit from the existence of low-priced imports. Granted, this competition can put some domestic producers out of business, but it's important to remember that consumers benefit more than producers lose in exactly the same way as when other countries are playing "fair" but happen to be able to produce at lower cost anyway.
In summary, the typical arguments made against free trade are generally not convincing enough to outweigh the benefits of free trade except in very particular circumstances.