Jumping an international border should be seen as a last resort. Not only is it illegal, but it is also extremely dangerous and can land you in prison indefinitely. More, jumping a border is no easy endeavor. It requires a great deal of preparation and not to mention nerve to pull it off. So unless you have dedicated yourself to becoming a long term and stateless penniless traveler or really need to leave or enter a country undetected, it is highly unadvisable.
Successfully jumping a border greatly depends on the border you are trying to jump. For example, since the completion of the Israeli fence, the border between Egypt and Israel has become nearly impossible to jump. However, most borders, simply due to their length or lack of funding, do not have such high security measures. This includes one very popular border crossing, the border between Turkey and Greece/Bulgaria. Since it is the most popular and one that I have had direct experience with, we will use this as an example.
The first step in jumping this border is the preparation process. This includes scouting the border crossing, familiarizing oneself with the terrain using google maps and collecting the necessary tools. You must first decide on the location where you will make the jump. The best place to do this is generally a few kilometers away from the official crossing. It is preferable a location where there is a good deal of foliage and dense trees to help obscure your activities. A good time to make the jump is in the cover of darkness between 4 and 5 am.
Once you have prepared and feel confident in the location, it is time to make the jump. In order to be successful, you must fully commit to what you are going to do. Since most borders in the world will have some kind of barbwire fence, this will be your main obstacle. There are two ways to overcome these fences. The first is to climb it. However, this can often leave one considerably cut. The second method is to try and pry the wires apart and slip through it. To achieve this, either wrap your shirt or a rope around the two wires. Find a sturdy stick nearby and use it to twist the rope. The wires should eventually part allowing enough space for you to slip through. A second common obstacle is a river. Some body of water demarcates almost all the borders of the world. To overcome this, your going to have to swim.
Once you make it to the next country, try to make as quickly and stealthily away from the border as fast as you can. Do not let local farmers or townspeople spot you as they will be sure to call the police. Move during the night and conceal yourself to sleep during the day. Do this until you have moved sufficiently far enough away from the border.
Although it is possible to jump a border, it comes with great risk. Unless you have no choice but to do so, it is not recommended. Some other things to note, if you enter a country without a visa this means that when you leave, you have to jump the border once again. You can try and turn up at your embassy claiming a lost passport, but with all the computer databases, they might also see that you entered the country illegally. In the end, jumping a border to save a few dollars on a visa is just not worth it.