Representatives of the US government and the EU have been looking towards breaking down barriers between their two markets for both goods and investments, with the eventual aim of making a free-trade agreement between the two akin to NAFTA that is across North America. This has also driven Canada to seek agreements with the EU and EFTA (Iceland, Norway, Switzerland, and Liechtenstein), while Mexico has already negotiated free trade agreements with both entities.
There is also the concern from nations who have trade agreements with the EU, such as Turkey, who will now be open to American goods without a reciprocating agreement for Turkish goods in the American market.
This was not the first time the agreement was suggested. Back in the mid-90s there was a similar suggestion pushed by the US to ensure that ties between the US and Europe remain strong in the post-Cold War world, but this stalled with the unpopularity of NAFTA taking hold in the US as domestic industries were whittled down. This agreement had been floated since then but was not seriously considered again until late last year, shortly after President Obama pledged to lower trade barriers in major markets for US businesses.
Looking at this current episode, "TAFTA" isn't being discussed much in the US but it is getting more attention in some EU member states, particularly among the farming sector who worry about American agricultural goods overwhelming them. Both the US and EU have protections on their agricultural goods, and it'll likely be one area that will be discussed to lower the trade barriers there or eliminate it all together- something that'd benefit the US more as farmers say. As such opposition in Europe has mostly taken the form of farmers protesting it in agricultural-heavy countries like France and Germany, often with the local green parties in those countries working alongside them taking the angle that US agricultural goods are not clean and under the same restrictions in the European market
. There is also the issue of NSA activities in European governments which has added skepticism among some people about the US's intentions for this agreement.
Some people in the media and tech end of things are alleging that there'll be provisions for overreaching IP protection like with ACTA
which the EU basically torpedoed after too much public attention came on it. In combination with the US also pursuing the Trans-Pacific Partnership
for its interests in the pacific countries with similar concerns of IP provisions, there is worry of a double whammy hitting if both go into full effect.