As Seth stated thread safe means that a method or class instance can be used by multiple threads at the same time without any problems occuring.
Consider the following method:
private int myInt = 0;
public int AddOne()
{
int tmp = myInt;
tmp = tmp + 1;
myInt = tmp;
return tmp;
}
Now thread A and thread B both would like to execute AddOne(). but A starts first and reads the value of myInt (0) into tmp. Now for some reason the scheduler decides to halt thread A and defer execution to thread B. Thread B now also reads the value of myInt (still 0) into it's own variable tmp. Thread B finishes the entire method, so in the end myInt = 1. And 1 is returned. Now it's Thread A's turn again. Thread A continues. And adds 1 to tmp (tmp was 0 for thread A). And then saves this value in myInt. myInt is again 1.
So in this case the method AddOne() was called two times, but because the method was not implemented in a thread safe way the value of myInt is not 2, as expected, but 1 because the second thread read the variable myInt before the first thread finished updating it.
Creating thread safe methods is very hard in non trivial cases. And there are quite a few techniques. In Java you can mark a method as synchronized, this means that only one thread can execute that method at a given time. The other threads wait in line. This makes a method thread safe, but if there is a lot of work to be done in a method, then this wastes a lot of time. Another technique is to 'mark only a small part of a method as synchronized' by creating a lock or semaphore, and locking this small part (usually called the critical section). There are even some methods that are implemented as lockless thread safe, which means that they are built in such a way that multiple threads can race through them at the same time without ever causing problems, this can be the case when a method only executes one atomic call. Atomic calls are calls that can't be interrupted and can only be done by one thread at a time.