Other answers have pointed out that the employer and employee can both be sued, and the employer is more likely to be insured, or able to pay, for damages. But why would the employer pay to insure the liabilities of its employees? What happens if a negligent employee is wealthy and the employer company is insolvent?

The plaintiff can recover solely from the employee. This follows from the other answers. But the employee may also be able to recover contribution or an indemnity from the employer. Conversely, an employer may seek contribution from a wealthy employee or well-resourced subcontractor. Because an employer may be partly or wholly liable for its employees' negligence, even if the plaintiff chooses not to sue the employer, it is rational for the employer to insure this risk.

Whether contribution and indemnity are available depends on your jurisdiction, but you can get a high-level view of the general principles in the US and UK by reading old American law review articles, like Bohlen, 'Contribution and indemnity between tortfeasors' (1936) 21:4 Cornell Law Review 552. It would be better to look at a textbook or cases from your jurisdiction, but they usually cost money. According to Bohlen, writing in 1936:

The rule which, except as modified by statute, is accepted law in every common-law jurisdiction other than Minnesota, Pennsylvania, Wisconsin, and, probably, Oregon is that there can be no contribution between joint tortfeasors ... the idea that the burden of bearing the loss caused by the tortious misconduct of two or more persons should not be determined by the choice of the injured person, but by a fair distribution among all those responsible for it, has led not only to the enactment of more or less, but generally less, effective acts allowing contribution in England, in many of the Canadian provinces, and in twelve American states, but also to an express rejection of the rule in Merryweather v. Nixan [(1799) 88 TR 186; 101 ER 1337] by direct decision in Minnesota, Pennsylvania, and Wisconsin, and by a strong dictum in Oregon.

Contribution refers to the right of a joint tortfeasor (somebody who is jointly liable with another for a tort such as negligence, such as the employee and employer) to recover a financial contribution from a co-tortfeasor. Indemnity simply refers to the right to claim 100% contribution from the co-tortfeasor. Bohlen explains why the doctrine of contribution breaks with the history of common law torts:

... the denial of contribution and the concept of tort liability upon which that denial is itself founded have greatly handicapped the intelligent development of the law of torts, and particularly the law of negligence. Had the right to contribution been recognized, particularly had it been made effective through a process by which the one of two co-tortfeasors against whom a separate action was brought could bring the other into court as a co-defendant, the burden of bearing the cost of the liability to which they were both subject could have been fairly divided between them. Had this been done, it would have been more than possible that the instinctive reaction of the bulk of mankind, that in some way or other the later of two wrongdoers is the more responsible for an accident of which both wrongdoings are a substantial cause, would have found a proper expression by requiring the later of the wrongdoers to pay a larger part of the damages than the earlier. Not only would all the intricacies and uncertainties of what the English lawyers call the doctrine of "novus actus interveniens" have been avoided, but a far more equitable result would have been attained.

It is a principle in equity and admiralty and, according to Bohlen, of the common law 'that those who stand in equal risk should bear the burden equally.' So, in many jurisdictions, there is a statute which prescribes a method of adjusting liability between joint tortfeasors whether or not all are sued by the injured party. Local civil procedure may allow defendants to join co-tortfeasors as defendants or third parties to the principal action.

###Is it true that someone (even if low level), similarly needs to have "standing" (or its equivalent) to be a defendant, and if so, what constitutes such "standing?"

Generally, a defendant does not need to have standing. However, a civil claim must plead a cause of action against each defendant. For example, an employee might be named as a defendant because he or she was personally negligent, and the employer might be named on the basis of its vicarious liability for the employee's negligence. If no cause of action exists, even assuming the facts asserted by the plaintiff, then the defendant may be able to demur to the claim or seek summary judgment, depending on the relevant court's civil procedure.