When the Supreme Court has an even number of seats, that sometimes means that their decisions on certain cases are tied, and that no side has a majority. In this circumstance, the Court can do a couple of things.
First, the Court can choose to hold cases over for reargument. If a majority of the Justices on the Supreme Court vote to hold a case over for reargument, then they will re-hear the case next term and decide on it then. Ideally, in the next term, the Court will have all nine Justices serving on the bench. However, if the Senate fails to hear or confirm a nominee this year, as some of its leading members have promised, then depending on how long the nomination process takes next year, the Court may have to hold cases over for reargument twice. The Supreme Court has held a number of high profile cases over for reargument due to vacancies, including Brown v. Board of Education and Roe v. Wade.
Second, if the Court chooses not to hold a case over for reargument, but still fails to reach any majority decision, then the lower court's decision is affirmed by an evenly divided vote. In this circumstance, the decision of the lower court stands, but does not set a national precedent. The Supreme Court may be okay doing this if there is no division among the lower courts; however, if the lower courts disagree as to a question of law, the Supreme Court will likely seek to resolve the disagreement by holding a case over for reargument. This is done in order to avoid a situation in which the Constitution means different things in different parts of the country.
Congress could, of course, change the number of seats in the Supreme Court. Were they to fix it permanently at eight, the Supreme Court would gain nothing by holding cases over for reargument. However, it is unlikely that this would happen. There are a number of cases likely to be decided by narrow margins in the Supreme Court arising from divisions among the lower courts (including cases this term like Zubik v. Burwell), so legislation that leaves these divisions intact could cause problems.
Furthermore, the last plan to change the number of seats on the Supreme Court (Franklin D. Roosevelt's infamous "court packing" plan) was vehemently rejected; the number of seats on the Supreme Court has remained at 9 for most of the nation's history, so there's no reason to think that a single contentious nomination would cause that to change.