MeSH stands for Medical Subject Headings and is a thesaurus, a vocabulary for the purpose of indexing journal articles and books. You can use MeSH to search (in PubMed/MEDLINE) for articles regarding a particular topic or condition. Also, there is a hierarchical structure, meaning that MeSH terms are organized in a tree structure. If you use a particular MeSH (say, "vascular disease") you will also find articles with related terms (say, "coronary heart disease", "myocardial infarction" and so on).
I am less familiar with Index Medicus, but the clearest definition I can find is that it is a comprehensive bibliographic index of scientific journal articles focusing on medical science fields. It contains journal titles and abbreviations.