As dismalscience said, none of these terms has a truly rigorous and concrete definition. But here is a broad outline:
Environmental economics is the branch of economics that deals with environmental issues, such as providing incentives for pollution abatement, dealling with climate change, managing shared environmental/natural resources, etc.
Neoclassical economics is the dominant paradigm for economic analysis across virtually all fields of economics (including environmental econ). It is cheifly characterised by the assumptions that all actors behave rationally in the pursuit of some well-defined objective. In modern economic work, neoclassical analysis typically manifests in the presentation of a mathematical model of behaviour that embodies this rationality assumption.
Welfare economics is the branch of economics that deals with questions about how the well being of individuals/groups in society (as well as of society in aggregate) can be definied/measured. It also deals with studying the effects of particular policies/economic phenomena on welfare. Such analysis is usually conducted within the framework of neoclassical economics.
Welfare economics provides an important set of tools used by environmental economists to understand who wins and loses under different regimes. It also provides the tools necessary to define what a "good" outcome might be, when all outcomes involve some winners and some losers.
Institutional economics studies how and why particular forms of institutions emerge and how they affect economic outcomes. One particular focus of institutional economics is how society deals with situations where markets would otherwise fail. For example, the environment is a common resource shared by everyone, meaning that individual actors have an incentive to exploit the resource for private gains and hope that others will take care of it instead. A key question addressed by institutional economics is how institutions have evolved to deal with these kinds of problems. Such analysis sometimes, but not always, occurs within the framework of neoclassical economics.